1.309 problem 310

Internal problem ID [8646]
Internal file name [OUTPUT/7579_Sunday_June_05_2022_11_07_52_PM_26852208/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 310.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "exact", "dAlembert", "homogeneousTypeD2", "first_order_ode_lie_symmetry_calculated"

Maple gives the following as the ode type

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\[ \boxed {\left (2 y^{3}+5 y x^{2}\right ) y^{\prime }+5 y^{2} x=-x^{3}} \]

1.309.1 Solving as homogeneousTypeD2 ode

Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} \left (2 u \left (x \right )^{3} x^{3}+5 u \left (x \right ) x^{3}\right ) \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right )+5 u \left (x \right )^{2} x^{3} = -x^{3} \end {align*}

In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {2 u^{4}+10 u^{2}+1}{x \left (2 u^{3}+5 u \right )} \end {align*}

Where \(f(x)=-\frac {1}{x}\) and \(g(u)=\frac {2 u^{4}+10 u^{2}+1}{2 u^{3}+5 u}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {2 u^{4}+10 u^{2}+1}{2 u^{3}+5 u}} \,du &= -\frac {1}{x} \,d x \\ \int { \frac {1}{\frac {2 u^{4}+10 u^{2}+1}{2 u^{3}+5 u}} \,du} &= \int {-\frac {1}{x} \,d x} \\ \frac {\ln \left (u^{4}+5 u^{2}+\frac {1}{2}\right )}{4}&=-\ln \left (x \right )+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} \frac {\sqrt {2}\, \left (4 u^{4}+20 u^{2}+2\right )^{{1}/{4}}}{2} &= {\mathrm e}^{-\ln \left (x \right )+c_{2}} \end {align*}

Which simplifies to \begin {align*} \frac {\sqrt {2}\, \left (4 u^{4}+20 u^{2}+2\right )^{{1}/{4}}}{2} &= \frac {c_{3}}{x} \end {align*}

Which simplifies to \[ \frac {\sqrt {2}\, \left (4 u \left (x \right )^{4}+20 u \left (x \right )^{2}+2\right )^{{1}/{4}}}{2} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] The solution is \[ \frac {\sqrt {2}\, \left (4 u \left (x \right )^{4}+20 u \left (x \right )^{2}+2\right )^{{1}/{4}}}{2} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \frac {\sqrt {2}\, \left (\frac {4 y^{4}}{x^{4}}+\frac {20 y^{2}}{x^{2}}+2\right )^{{1}/{4}}}{2} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x}\\ \frac {2^{{3}/{4}} \left (\frac {2 y^{4}+10 y^{2} x^{2}+x^{4}}{x^{4}}\right )^{{1}/{4}}}{2} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {2^{{3}/{4}} \left (\frac {2 y^{4}+10 y^{2} x^{2}+x^{4}}{x^{4}}\right )^{{1}/{4}}}{2} &= \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \\ \end{align*}

Figure 402: Slope field plot

Verification of solutions

\[ \frac {2^{{3}/{4}} \left (\frac {2 y^{4}+10 y^{2} x^{2}+x^{4}}{x^{4}}\right )^{{1}/{4}}}{2} = \frac {c_{3} {\mathrm e}^{c_{2}}}{x} \] Verified OK.

1.309.2 Solving as first order ode lie symmetry calculated ode

Writing the ode as \begin {align*} y^{\prime }&=-\frac {x \left (x^{2}+5 y^{2}\right )}{y \left (5 x^{2}+2 y^{2}\right )}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}

The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}

The type of this ode is not in the lookup table. To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} b_{2}-\frac {x \left (x^{2}+5 y^{2}\right ) \left (b_{3}-a_{2}\right )}{y \left (5 x^{2}+2 y^{2}\right )}-\frac {x^{2} \left (x^{2}+5 y^{2}\right )^{2} a_{3}}{y^{2} \left (5 x^{2}+2 y^{2}\right )^{2}}-\left (-\frac {x^{2}+5 y^{2}}{y \left (5 x^{2}+2 y^{2}\right )}-\frac {2 x^{2}}{y \left (5 x^{2}+2 y^{2}\right )}+\frac {10 x^{2} \left (x^{2}+5 y^{2}\right )}{y \left (5 x^{2}+2 y^{2}\right )^{2}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (-\frac {10 x}{5 x^{2}+2 y^{2}}+\frac {x \left (x^{2}+5 y^{2}\right )}{y^{2} \left (5 x^{2}+2 y^{2}\right )}+\frac {4 x \left (x^{2}+5 y^{2}\right )}{\left (5 x^{2}+2 y^{2}\right )^{2}}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ -\frac {x^{6} a_{3}+5 x^{6} b_{2}-10 x^{5} y a_{2}+10 x^{5} y b_{3}+5 x^{4} y^{2} a_{3}-44 x^{4} y^{2} b_{2}-8 x^{3} y^{3} a_{2}+8 x^{3} y^{3} b_{3}+44 x^{2} y^{4} a_{3}-10 x^{2} y^{4} b_{2}-20 x \,y^{5} a_{2}+20 x \,y^{5} b_{3}-10 y^{6} a_{3}-4 y^{6} b_{2}+5 x^{5} b_{1}-5 x^{4} y a_{1}-19 x^{3} y^{2} b_{1}+19 x^{2} y^{3} a_{1}+10 x \,y^{4} b_{1}-10 y^{5} a_{1}}{y^{2} \left (5 x^{2}+2 y^{2}\right )^{2}} = 0 \] Setting the numerator to zero gives \begin{equation} \tag{6E} -x^{6} a_{3}-5 x^{6} b_{2}+10 x^{5} y a_{2}-10 x^{5} y b_{3}-5 x^{4} y^{2} a_{3}+44 x^{4} y^{2} b_{2}+8 x^{3} y^{3} a_{2}-8 x^{3} y^{3} b_{3}-44 x^{2} y^{4} a_{3}+10 x^{2} y^{4} b_{2}+20 x \,y^{5} a_{2}-20 x \,y^{5} b_{3}+10 y^{6} a_{3}+4 y^{6} b_{2}-5 x^{5} b_{1}+5 x^{4} y a_{1}+19 x^{3} y^{2} b_{1}-19 x^{2} y^{3} a_{1}-10 x \,y^{4} b_{1}+10 y^{5} a_{1} = 0 \end{equation} Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \{x, y\} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \{x = v_{1}, y = v_{2}\} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} 10 a_{2} v_{1}^{5} v_{2}+8 a_{2} v_{1}^{3} v_{2}^{3}+20 a_{2} v_{1} v_{2}^{5}-a_{3} v_{1}^{6}-5 a_{3} v_{1}^{4} v_{2}^{2}-44 a_{3} v_{1}^{2} v_{2}^{4}+10 a_{3} v_{2}^{6}-5 b_{2} v_{1}^{6}+44 b_{2} v_{1}^{4} v_{2}^{2}+10 b_{2} v_{1}^{2} v_{2}^{4}+4 b_{2} v_{2}^{6}-10 b_{3} v_{1}^{5} v_{2}-8 b_{3} v_{1}^{3} v_{2}^{3}-20 b_{3} v_{1} v_{2}^{5}+5 a_{1} v_{1}^{4} v_{2}-19 a_{1} v_{1}^{2} v_{2}^{3}+10 a_{1} v_{2}^{5}-5 b_{1} v_{1}^{5}+19 b_{1} v_{1}^{3} v_{2}^{2}-10 b_{1} v_{1} v_{2}^{4} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (-a_{3}-5 b_{2}\right ) v_{1}^{6}+\left (10 a_{2}-10 b_{3}\right ) v_{1}^{5} v_{2}-5 b_{1} v_{1}^{5}+\left (-5 a_{3}+44 b_{2}\right ) v_{1}^{4} v_{2}^{2}+5 a_{1} v_{1}^{4} v_{2}+\left (8 a_{2}-8 b_{3}\right ) v_{1}^{3} v_{2}^{3}+19 b_{1} v_{1}^{3} v_{2}^{2}+\left (-44 a_{3}+10 b_{2}\right ) v_{1}^{2} v_{2}^{4}-19 a_{1} v_{1}^{2} v_{2}^{3}+\left (20 a_{2}-20 b_{3}\right ) v_{1} v_{2}^{5}-10 b_{1} v_{1} v_{2}^{4}+\left (10 a_{3}+4 b_{2}\right ) v_{2}^{6}+10 a_{1} v_{2}^{5} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} -19 a_{1}&=0\\ 5 a_{1}&=0\\ 10 a_{1}&=0\\ -10 b_{1}&=0\\ -5 b_{1}&=0\\ 19 b_{1}&=0\\ 8 a_{2}-8 b_{3}&=0\\ 10 a_{2}-10 b_{3}&=0\\ 20 a_{2}-20 b_{3}&=0\\ -44 a_{3}+10 b_{2}&=0\\ -5 a_{3}+44 b_{2}&=0\\ -a_{3}-5 b_{2}&=0\\ 10 a_{3}+4 b_{2}&=0 \end {align*}

Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=b_{3}\\ a_{3}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=b_{3} \end {align*}

Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives \begin{align*} \xi &= x \\ \eta &= y \\ \end{align*} Shifting is now applied to make \(\xi =0\) in order to simplify the rest of the computation \begin {align*} \eta &= \eta - \omega \left (x,y\right ) \xi \\ &= y - \left (-\frac {x \left (x^{2}+5 y^{2}\right )}{y \left (5 x^{2}+2 y^{2}\right )}\right ) \left (x\right ) \\ &= \frac {x^{4}+10 y^{2} x^{2}+2 y^{4}}{5 x^{2} y +2 y^{3}}\\ \xi &= 0 \end {align*}

The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.

The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}

The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Since \(\xi =0\) then in this special case \begin {align*} R = x \end {align*}

\(S\) is found from \begin {align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{\frac {x^{4}+10 y^{2} x^{2}+2 y^{4}}{5 x^{2} y +2 y^{3}}}} dy \end {align*}

Which results in \begin {align*} S&= \frac {\ln \left (x^{4}+10 y^{2} x^{2}+2 y^{4}\right )}{4} \end {align*}

Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating \begin {align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end {align*}

Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode given by \begin {align*} \omega (x,y) &= -\frac {x \left (x^{2}+5 y^{2}\right )}{y \left (5 x^{2}+2 y^{2}\right )} \end {align*}

Evaluating all the partial derivatives gives \begin {align*} R_{x} &= 1\\ R_{y} &= 0\\ S_{x} &= \frac {x \left (x^{2}+5 y^{2}\right )}{x^{4}+10 y^{2} x^{2}+2 y^{4}}\\ S_{y} &= \frac {5 x^{2} y +2 y^{3}}{x^{4}+10 y^{2} x^{2}+2 y^{4}} \end {align*}

Substituting all the above in (2) and simplifying gives the ode in canonical coordinates. \begin {align*} \frac {dS}{dR} &= 0\tag {2A} \end {align*}

We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives \begin {align*} \frac {dS}{dR} &= 0 \end {align*}

The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\). Integrating the above gives \begin {align*} S \left (R \right ) = c_{1}\tag {4} \end {align*}

To complete the solution, we just need to transform (4) back to \(x,y\) coordinates. This results in \begin {align*} \frac {\ln \left (2 y^{4}+10 y^{2} x^{2}+x^{4}\right )}{4} = c_{1} \end {align*}

Which simplifies to \begin {align*} \frac {\ln \left (2 y^{4}+10 y^{2} x^{2}+x^{4}\right )}{4} = c_{1} \end {align*}

The following diagram shows solution curves of the original ode and how they transform in the canonical coordinates space using the mapping shown.

Original ode in \(x,y\) coordinates	Canonical coordinates transformation	ODE in canonical coordinates \((R,S)\)
\( \frac {dy}{dx} = -\frac {x \left (x^{2}+5 y^{2}\right )}{y \left (5 x^{2}+2 y^{2}\right )}\)		\( \frac {d S}{d R} = 0\)
	\(\!\begin {aligned} R&= x\\ S&= \frac {\ln \left (x^{4}+10 y^{2} x^{2}+2 y^{4}\right )}{4} \end {aligned} \)

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {\ln \left (2 y^{4}+10 y^{2} x^{2}+x^{4}\right )}{4} &= c_{1} \\ \end{align*}

Figure 403: Slope field plot

Verification of solutions

\[ \frac {\ln \left (2 y^{4}+10 y^{2} x^{2}+x^{4}\right )}{4} = c_{1} \] Verified OK.

1.309.3 Solving as exact ode

Entering Exact first order ODE solver. (Form one type)

To solve an ode of the form\begin {equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A} \end {equation} We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \] Hence\begin {equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B} \end {equation} Comparing (A,B) shows that\begin {align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end {align*}

But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\] If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is \[ M(x,y) \mathop {\mathrm {d}x}+ N(x,y) \mathop {\mathrm {d}y}=0 \tag {1A} \] Therefore \begin {align*} \left (5 x^{2} y +2 y^{3}\right )\mathop {\mathrm {d}y} &= \left (-x^{3}-5 y^{2} x\right )\mathop {\mathrm {d}x}\\ \left (x^{3}+5 y^{2} x\right )\mathop {\mathrm {d}x} + \left (5 x^{2} y +2 y^{3}\right )\mathop {\mathrm {d}y} &= 0 \tag {2A} \end {align*}

Comparing (1A) and (2A) shows that \begin {align*} M(x,y) &= x^{3}+5 y^{2} x\\ N(x,y) &= 5 x^{2} y +2 y^{3} \end {align*}

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied \[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial x} \] Using result found above gives \begin {align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (x^{3}+5 y^{2} x\right )\\ &= 10 x y \end {align*}

And \begin {align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (5 x^{2} y +2 y^{3}\right )\\ &= 10 x y \end {align*}

Since \(\frac {\partial M}{\partial y}= \frac {\partial N}{\partial x}\), then the ODE is exact The following equations are now set up to solve for the function \(\phi \left (x,y\right )\) \begin {align*} \frac {\partial \phi }{\partial x } &= M\tag {1} \\ \frac {\partial \phi }{\partial y } &= N\tag {2} \end {align*}

Integrating (1) w.r.t. \(x\) gives \begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int M\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int x^{3}+5 y^{2} x\mathop {\mathrm {d}x} \\ \tag{3} \phi &= \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}+ f(y) \\ \end{align*} Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives \begin{equation} \tag{4} \frac {\partial \phi }{\partial y} = 5 \left (x^{2}+5 y^{2}\right ) y+f'(y) \end{equation} But equation (2) says that \(\frac {\partial \phi }{\partial y} = 5 x^{2} y +2 y^{3}\). Therefore equation (4) becomes \begin{equation} \tag{5} 5 x^{2} y +2 y^{3} = 5 \left (x^{2}+5 y^{2}\right ) y+f'(y) \end{equation} Solving equation (5) for \( f'(y)\) gives \[ f'(y) = -23 y^{3} \] Integrating the above w.r.t \(y\) gives \begin{align*} \int f'(y) \mathop {\mathrm {d}y} &= \int \left ( -23 y^{3}\right ) \mathop {\mathrm {d}y} \\ f(y) &= -\frac {23 y^{4}}{4}+ c_{1} \\ \end{align*} Where \(c_{1}\) is constant of integration. Substituting result found above for \(f(y)\) into equation (3) gives \(\phi \) \[ \phi = \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4}+ c_{1} \] But since \(\phi \) itself is a constant function, then let \(\phi =c_{2}\) where \(c_{2}\) is new constant and combining \(c_{1}\) and \(c_{2}\) constants into new constant \(c_{1}\) gives the solution as \[ c_{1} = \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} &= c_{1} \\ \end{align*}

Figure 404: Slope field plot

Verification of solutions

\[ \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} = c_{1} \] Verified OK.

1.309.4 Solving as dAlembert ode

Let \(p=y^{\prime }\) the ode becomes \begin {align*} \left (5 x^{2} y +2 y^{3}\right ) p +5 y^{2} x = -x^{3} \end {align*}

Solving for \(y\) from the above results in \begin {align*} y &= \left (\frac {\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}{6 p}-\frac {5 \left (6 p^{2}-5\right )}{6 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {5}{6 p}\right ) x\tag {1A}\\ y &= \left (-\frac {\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}{12 p}+\frac {\frac {5 p^{2}}{2}-\frac {25}{12}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {5}{6 p}+\frac {i \sqrt {3}\, \left (\frac {\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}{6 p}+\frac {5 p^{2}-\frac {25}{6}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}\right )}{2}\right ) x\tag {2A}\\ y &= \left (-\frac {\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}{12 p}+\frac {\frac {5 p^{2}}{2}-\frac {25}{12}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {5}{6 p}-\frac {i \sqrt {3}\, \left (\frac {\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}{6 p}+\frac {5 p^{2}-\frac {25}{6}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}\right )}{2}\right ) x\tag {3A} \end {align*}

This has the form \begin {align*} y=xf(p)+g(p)\tag {*} \end {align*}

Where \(f,g\) are functions of \(p=y'(x)\). Each of the above ode’s is dAlembert ode which is now solved. Solving ode 1A Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}

Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= \frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}{6 p}\\ g &= 0 \end {align*}

Hence (2) becomes \begin {align*} p -\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}{6 p} = x \left (-\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}{6 p^{2}}+\frac {\frac {\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}}-\frac {60 p}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+\frac {5 \left (6 p^{2}-5\right ) \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}}}}{6 p}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} p -\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}{6 p} = 0 \end {align*}

Solving for \(p\) from the above gives \begin {align*} p&=\frac {\sqrt {-10+2 \sqrt {23}}}{2}\\ p&=\frac {\sqrt {-10-2 \sqrt {23}}}{2}\\ p&=-\frac {\sqrt {-10+2 \sqrt {23}}}{2}\\ p&=-\frac {\sqrt {-10-2 \sqrt {23}}}{2} \end {align*}

Substituting these in (1A) gives \begin {align*} y&=\frac {-x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+10 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{6 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-10 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+60 x \sqrt {23}+400 x}{6 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {-x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+10 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{6 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-10 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{6 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \end {align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} p^{\prime }\left (x \right ) = \frac {p \left (x \right )-\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p \left (x \right )^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{1}/{3}}}}{6 p \left (x \right )}}{x \left (-\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p \left (x \right )^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{1}/{3}}}}{6 p \left (x \right )^{2}}+\frac {\frac {\frac {3 \sqrt {3}\, p \left (x \right ) \left (4000 p \left (x \right )^{3}-2834 p \left (x \right )\right )}{2 \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}+342 p \left (x \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{2}/{3}}}-\frac {60 p \left (x \right )}{\left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{1}/{3}}}+\frac {5 \left (6 p \left (x \right )^{2}-5\right ) \left (\frac {3 \sqrt {3}\, p \left (x \right ) \left (4000 p \left (x \right )^{3}-2834 p \left (x \right )\right )}{2 \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}+342 p \left (x \right )\right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p \left (x \right )^{4}-1417 p \left (x \right )^{2}+500}\, p \left (x \right )+171 p \left (x \right )^{2}-125\right )^{{4}/{3}}}}{6 p \left (x \right )}\right )}\tag {3} \end {align*}

This ODE is now solved for \(p \left (x \right )\).

Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = \frac {x \left (p \right ) \left (-\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}{6 p^{2}}+\frac {\frac {\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}}-\frac {60 p}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+\frac {5 \left (6 p^{2}-5\right ) \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}}}}{6 p}\right )}{p -\frac {-5+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-\frac {5 \left (6 p^{2}-5\right )}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}{6 p}}\tag {4} \end {align*}

This ODE is now solved for \(x \left (p \right )\).

Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}

Where here \begin {align*} p(p) &=-\frac {-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90000 p^{7} \sqrt {3}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}\\ q(p) &=0 \end {align*}

Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )-\frac {x \left (p \right ) \left (-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90000 p^{7} \sqrt {3}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}\right )}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )} = 0 \end {align*}

The integrating factor \(\mu \) is \[ \mu = {\mathrm e}^{\int -\frac {-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90000 p^{7} \sqrt {3}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \mu x &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left ({\mathrm e}^{\int -\frac {-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90000 p^{7} \sqrt {3}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} x\right ) &= 0 \end {align*}

Integrating gives \begin {align*} {\mathrm e}^{\int -\frac {-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90000 p^{7} \sqrt {3}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} x &= c_{2} \end {align*}

Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\int -\frac {-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90000 p^{7} \sqrt {3}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p}\) results in \begin {align*} x \left (p \right ) &= c_{2} {\mathrm e}^{-\left (\int \frac {15000 \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}} p^{5}-107490 p^{5} \sqrt {3}+1417 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-21255 \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}} p^{3}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-57 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+147125 p^{3} \sqrt {3}-1000 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+855 \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}} p^{2}+7500 \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}} p -3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+125 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-50000 p \sqrt {3}-625 \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+3125 \sqrt {1000 p^{4}-1417 p^{2}+500}}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (-6 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}-5 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p \right )} \end {align*}

Since the solution \(x \left (p \right )\) has unresolved integral, unable to continue.

Solving ode 2A Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}

Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= \frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}\\ g &= 0 \end {align*}

Hence (2) becomes \begin {align*} p -\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}} = x \left (\frac {\frac {2 i \sqrt {3}\, \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+60 i \sqrt {3}\, p -\frac {2 \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+60 p -\frac {10 \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}}}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25}{12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {\left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right ) \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{36 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}}}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} p -\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}} = 0 \end {align*}

Solving for \(p\) from the above gives \begin {align*} p&=\frac {\sqrt {-10+2 \sqrt {23}}}{2}\\ p&=\frac {\sqrt {-10-2 \sqrt {23}}}{2}\\ p&=-\frac {\sqrt {-10+2 \sqrt {23}}}{2}\\ p&=-\frac {\sqrt {-10-2 \sqrt {23}}}{2} \end {align*}

Substituting these in (1A) gives \begin {align*} y&=\frac {-i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}+x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+20 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}-x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-20 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {-i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}+x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+20 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}-x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}-20 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \end {align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} \text {Expression too large to display}\tag {3} \end {align*}

This ODE is now solved for \(p \left (x \right )\).

Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = \frac {x \left (p \right ) \left (\frac {\frac {2 i \sqrt {3}\, \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+60 i \sqrt {3}\, p -\frac {2 \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+60 p -\frac {10 \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}}}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25}{12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-\frac {\left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right ) \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{36 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}}}\right )}{p -\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}\tag {4} \end {align*}

This ODE is now solved for \(x \left (p \right )\).

Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}

Where here \begin {align*} p(p) &=\frac {-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-90000 p^{7} \sqrt {3}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right )}\\ q(p) &=0 \end {align*}

Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )+\frac {x \left (p \right ) \left (-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-90000 p^{7} \sqrt {3}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p \right )}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right )} = 0 \end {align*}

The integrating factor \(\mu \) is \[ \mu = {\mathrm e}^{\int \frac {-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-90000 p^{7} \sqrt {3}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right )}d p} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \mu x &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left ({\mathrm e}^{\int \frac {-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-90000 p^{7} \sqrt {3}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right )}d p} x\right ) &= 0 \end {align*}

Integrating gives \begin {align*} {\mathrm e}^{\int \frac {-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-90000 p^{7} \sqrt {3}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right )}d p} x &= c_{4} \end {align*}

Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\int \frac {-3125 \sqrt {1000 p^{4}-1417 p^{2}+500}-3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p -500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-90000 p^{7} \sqrt {3}+160020 p^{5} \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}-85850 p^{3} \sqrt {3}+3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 p^{2}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25\right )}d p}\) results in \begin {align*} x \left (p \right ) &= c_{4} {\mathrm e}^{\int \frac {-3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \left (19 \left (p^{2}-\frac {125}{57}\right ) \left (i \sqrt {3}-1\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}+1417 p \left (p^{2}-\frac {1000}{1417}\right ) \left (-i+\frac {\sqrt {3}}{3}\right )\right )+90 \left (\left (-\frac {125}{9}+19 p^{2}\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}+\frac {1000 p \left (p^{4}-\frac {1417}{1000} p^{2}+\frac {1}{2}\right ) \sqrt {3}}{3}\right ) \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+1710 \left (1+i \sqrt {3}\right ) \left (p^{4}+\frac {75}{38} p^{2}-\frac {625}{342}\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}+322470 \left (i+\frac {\sqrt {3}}{3}\right ) \left (p^{4}-\frac {29425}{21498} p^{2}+\frac {5000}{10749}\right ) p}{\sqrt {1000 p^{4}-1417 p^{2}+500}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \left (-i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} \end {align*}

Since the solution \(x \left (p \right )\) has unresolved integral, unable to continue.

Solving ode 3A Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}

Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= -\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}\\ g &= 0 \end {align*}

Hence (2) becomes \begin {align*} p +\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}} = x \left (-\frac {\frac {2 i \sqrt {3}\, \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+60 i \sqrt {3}\, p +\frac {\frac {\sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{\sqrt {1000 p^{4}-1417 p^{2}+500}}+2 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+228 p}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-60 p +\frac {\frac {5 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{\sqrt {1000 p^{4}-1417 p^{2}+500}}+10 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+1140 p}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}}}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25}{12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+\frac {\left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right ) \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{36 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}}}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} p +\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}} = 0 \end {align*}

Solving for \(p\) from the above gives \begin {align*} p&=\frac {\sqrt {-10+2 \sqrt {23}}}{2}\\ p&=\frac {\sqrt {-10-2 \sqrt {23}}}{2}\\ p&=-\frac {\sqrt {-10+2 \sqrt {23}}}{2}\\ p&=-\frac {\sqrt {-10-2 \sqrt {23}}}{2} \end {align*}

Substituting these in (1A) gives \begin {align*} y&=\frac {i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}+x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+20 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {-i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}-x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-20 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}+x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+20 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}}\\ y&=\frac {-i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}-x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}-20 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \end {align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} \text {Expression too large to display}\tag {3} \end {align*}

This ODE is now solved for \(p \left (x \right )\).

Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = \frac {x \left (p \right ) \left (-\frac {\frac {2 i \sqrt {3}\, \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{3 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+60 i \sqrt {3}\, p +\frac {\frac {\sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{\sqrt {1000 p^{4}-1417 p^{2}+500}}+2 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+228 p}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}-60 p +\frac {\frac {5 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{\sqrt {1000 p^{4}-1417 p^{2}+500}}+10 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+1140 p}{\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}}}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25}{12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}+\frac {\left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right ) \left (\frac {3 \sqrt {3}\, p \left (4000 p^{3}-2834 p \right )}{2 \sqrt {1000 p^{4}-1417 p^{2}+500}}+3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}+342 p \right )}{36 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}}}\right )}{p +\frac {i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25}{12 p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}}}\tag {4} \end {align*}

This ODE is now solved for \(x \left (p \right )\).

Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}

Where here \begin {align*} p(p) &=\frac {3125 \sqrt {1000 p^{4}-1417 p^{2}+500}+3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}-2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p +500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+90000 p^{7} \sqrt {3}-160020 p^{5} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+85850 p^{3} \sqrt {3}-3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}\\ q(p) &=0 \end {align*}

Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )+\frac {x \left (p \right ) \left (3125 \sqrt {1000 p^{4}-1417 p^{2}+500}+3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}-2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p +500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+90000 p^{7} \sqrt {3}-160020 p^{5} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+85850 p^{3} \sqrt {3}-3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p \right )}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )} = 0 \end {align*}

The integrating factor \(\mu \) is \[ \mu = {\mathrm e}^{\int \frac {3125 \sqrt {1000 p^{4}-1417 p^{2}+500}+3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}-2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p +500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+90000 p^{7} \sqrt {3}-160020 p^{5} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+85850 p^{3} \sqrt {3}-3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \mu x &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left ({\mathrm e}^{\int \frac {3125 \sqrt {1000 p^{4}-1417 p^{2}+500}+3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}-2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p +500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+90000 p^{7} \sqrt {3}-160020 p^{5} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+85850 p^{3} \sqrt {3}-3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} x\right ) &= 0 \end {align*}

Integrating gives \begin {align*} {\mathrm e}^{\int \frac {3125 \sqrt {1000 p^{4}-1417 p^{2}+500}+3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}-2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p +500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+90000 p^{7} \sqrt {3}-160020 p^{5} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+85850 p^{3} \sqrt {3}-3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} x &= c_{6} \end {align*}

Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\int \frac {3125 \sqrt {1000 p^{4}-1417 p^{2}+500}+3000 p^{5} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-90 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}-2834 p^{3} \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-75 \sqrt {3}\, \left (1000 p^{4}-1417 p^{2}+500\right ) p +500 \sqrt {3}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+114 i \sqrt {3}\, p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}+90000 p^{7} \sqrt {3}-160020 p^{5} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{4}/{3}} \sqrt {1000 p^{4}-1417 p^{2}+500}-1710 p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+85850 p^{3} \sqrt {3}-3375 p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}+270 i \left (1000 p^{4}-1417 p^{2}+500\right ) p^{3}+9000 i p^{5} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+225 i \left (1000 p^{4}-1417 p^{2}+500\right ) p -8502 i p^{3} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+1500 i p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-3125 i \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-12500 p \sqrt {3}+1710 i \sqrt {3}\, p^{4} \sqrt {1000 p^{4}-1417 p^{2}+500}+3375 i \sqrt {3}\, p^{2} \sqrt {1000 p^{4}-1417 p^{2}+500}-i \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{5}/{3}} \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}-270000 i p^{7}+480060 i p^{5}-257550 i p^{3}+37500 i p}{p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}\, \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p}\) results in \begin {align*} x \left (p \right ) &= c_{6} {\mathrm e}^{\int -\frac {322470 \left (\left (-\frac {19 \left (p^{2}-\frac {125}{57}\right ) \left (1+i \sqrt {3}\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}}{107490}+\frac {1417 \left (i+\frac {\sqrt {3}}{3}\right ) p \left (p^{2}-\frac {1000}{1417}\right )}{107490}\right ) \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+\left (\left (-\frac {19 p^{2}}{3583}+\frac {125}{32247}\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}-\frac {1000 p \left (p^{4}-\frac {1417}{1000} p^{2}+\frac {1}{2}\right ) \sqrt {3}}{10749}\right ) \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+\frac {19 \left (i \sqrt {3}-1\right ) \left (p^{4}+\frac {75}{38} p^{2}-\frac {625}{342}\right ) \sqrt {1000 p^{4}-1417 p^{2}+500}}{3583}+\left (p^{4}-\frac {29425}{21498} p^{2}+\frac {5000}{10749}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) p \right )}{\sqrt {1000 p^{4}-1417 p^{2}+500}\, p \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right ) \left (i \sqrt {3}\, \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}+30 i \sqrt {3}\, p^{2}+12 p^{2} \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}-25 i \sqrt {3}+\left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{2}/{3}}-30 p^{2}+10 \left (3 \sqrt {3}\, \sqrt {1000 p^{4}-1417 p^{2}+500}\, p +171 p^{2}-125\right )^{{1}/{3}}+25\right )}d p} \end {align*}

Since the solution \(x \left (p \right )\) has unresolved integral, unable to continue.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+10 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{6 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{2} y &= \frac {x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-10 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+60 x \sqrt {23}+400 x}{6 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{3} y &= \frac {-x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+10 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{6 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{4} y &= \frac {x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-10 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{6 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{5} y &= \frac {-i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}+x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+20 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{6} y &= \frac {i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}-x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-20 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{7} y &= \frac {-i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}+x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+20 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{8} y &= \frac {i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}-x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}-20 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{9} y &= \frac {i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}+x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+20 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{10} y &= \frac {-i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}-x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-20 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{11} y &= \frac {i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}+x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+20 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \\ \tag{12} y &= \frac {-i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}-x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}-20 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \\ \end{align*}

Figure 405: Slope field plot

Verification of solutions

\[ y = \frac {-x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+10 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{6 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-10 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+60 x \sqrt {23}+400 x}{6 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {-x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+10 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{6 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-10 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{6 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {-i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}+x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+20 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}-x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-20 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {-i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}+x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+20 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}-x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}-20 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}+x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}+20 x \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {-i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}-x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}-20 x \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10-2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420-684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {i x \sqrt {3}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 i x \sqrt {3}\, \sqrt {23}-400 i x \sqrt {3}+x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 x \sqrt {23}+20 x \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}+400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (828 i \sqrt {3}\, \sqrt {31}-180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

\[ y = \frac {-i x \sqrt {3}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}-60 i x \sqrt {3}\, \sqrt {23}+400 i x \sqrt {3}-x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{2}/{3}}+60 x \sqrt {23}-20 x \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}-400 x}{12 \sqrt {-10+2 \sqrt {23}}\, \left (-828 i \sqrt {3}\, \sqrt {31}+180 i \sqrt {3}\, \sqrt {23}\, \sqrt {31}-4420+684 \sqrt {23}\right )^{{1}/{3}}} \] Verified OK.

1.309.5 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (2 y^{3}+5 y x^{2}\right ) y^{\prime }+5 y^{2} x =-x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \square & {} & \textrm {Check if ODE is exact}\hspace {3pt} \\ {} & \circ & \textrm {ODE is exact if the lhs is the total derivative of a}\hspace {3pt} C^{2}\hspace {3pt}\textrm {function}\hspace {3pt} \\ {} & {} & F^{\prime }\left (x , y\right )=0 \\ {} & \circ & \textrm {Compute derivative of lhs}\hspace {3pt} \\ {} & {} & F^{\prime }\left (x , y\right )+\left (\frac {\partial }{\partial y}F \left (x , y\right )\right ) y^{\prime }=0 \\ {} & \circ & \textrm {Evaluate derivatives}\hspace {3pt} \\ {} & {} & 10 x y =10 x y \\ {} & \circ & \textrm {Condition met, ODE is exact}\hspace {3pt} \\ \bullet & {} & \textrm {Exact ODE implies solution will be of this form}\hspace {3pt} \\ {} & {} & \left [F \left (x , y\right )=c_{1} , M \left (x , y\right )=F^{\prime }\left (x , y\right ), N \left (x , y\right )=\frac {\partial }{\partial y}F \left (x , y\right )\right ] \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} F \left (x , y\right )\hspace {3pt}\textrm {by integrating}\hspace {3pt} M \left (x , y\right )\hspace {3pt}\textrm {with respect to}\hspace {3pt} x \\ {} & {} & F \left (x , y\right )=\int \left (x^{3}+5 y^{2} x \right )d x +\textit {\_F1} \left (y \right ) \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & F \left (x , y\right )=\frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}+\textit {\_F1} \left (y \right ) \\ \bullet & {} & \textrm {Take derivative of}\hspace {3pt} F \left (x , y\right )\hspace {3pt}\textrm {with respect to}\hspace {3pt} y \\ {} & {} & N \left (x , y\right )=\frac {\partial }{\partial y}F \left (x , y\right ) \\ \bullet & {} & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & 5 x^{2} y +2 y^{3}=5 \left (x^{2}+5 y^{2}\right ) y +\frac {d}{d y}\textit {\_F1} \left (y \right ) \\ \bullet & {} & \textrm {Isolate for}\hspace {3pt} \frac {d}{d y}\textit {\_F1} \left (y \right ) \\ {} & {} & \frac {d}{d y}\textit {\_F1} \left (y \right )=5 x^{2} y +2 y^{3}-5 \left (x^{2}+5 y^{2}\right ) y \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_F1} \left (y \right ) \\ {} & {} & \textit {\_F1} \left (y \right )=-\frac {23 y^{4}}{4} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_F1} \left (y \right )\hspace {3pt}\textrm {into equation for}\hspace {3pt} F \left (x , y\right ) \\ {} & {} & F \left (x , y\right )=\frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} F \left (x , y\right )\hspace {3pt}\textrm {into the solution of the ODE}\hspace {3pt} \\ {} & {} & \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4}=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {\sqrt {-10 x^{2}-2 \sqrt {23 x^{4}+8 c_{1}}}}{2}, y=\frac {\sqrt {-10 x^{2}-2 \sqrt {23 x^{4}+8 c_{1}}}}{2}, y=-\frac {\sqrt {-10 x^{2}+2 \sqrt {23 x^{4}+8 c_{1}}}}{2}, y=\frac {\sqrt {-10 x^{2}+2 \sqrt {23 x^{4}+8 c_{1}}}}{2}\right \} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying homogeneous D 
<- homogeneous successful`
 

✓ Solution by Maple

Time used: 0.094 (sec). Leaf size: 125

dsolve((2*y(x)^3+5*x^2*y(x))*diff(y(x),x)+5*x*y(x)^2+x^3 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-10 c_{1} x^{2}-2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {\sqrt {-10 c_{1} x^{2}-2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\sqrt {-10 c_{1} x^{2}+2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {\sqrt {-10 c_{1} x^{2}+2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 21.205 (sec). Leaf size: 295

DSolve[x^3 + 5*x*y[x]^2 + (5*x^2*y[x] + 2*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-5 x^2-\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-5 x^2-\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-5 x^2+\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-5 x^2+\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}} \\ \end{align*}