Internal problem ID [7369]
Internal file name [OUTPUT/6350_Sunday_June_05_2022_04_41_01_PM_39470954/index.tex]
Book: First order enumerated odes
Section: section 1
Problem number: 53.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "first_order_nonlinear_p_but_separable"
Maple gives the following as the ode type
[[_homogeneous, `class G`]]
\[ \boxed {{y^{\prime }}^{2}-\frac {y^{3}}{x}=0} \]
The ode has the form \begin {align*} (y')^{\frac {n}{m}} &= f(x) g(y)\tag {1} \end {align*}
Where \(n=2, m=1, f=\frac {1}{x} , g=y^{3}\). Hence the ode is \begin {align*} (y')^{2} &= \frac {y^{3}}{x} \end {align*}
Solving for \(y^{\prime }\) from (1) gives \begin {align*} y^{\prime } &=\sqrt {f g}\\ y^{\prime } &=-\sqrt {f g} \end {align*}
To be able to solve as separable ode, we have to now assume that \(f>0,g>0\). \begin {align*} \frac {1}{x} &> 0\\ y^{3} &> 0 \end {align*}
Under the above assumption the differential equations become separable and can be written as \begin {align*} y^{\prime } &=\sqrt {f}\, \sqrt {g}\\ y^{\prime } &=-\sqrt {f}\, \sqrt {g} \end {align*}
Therefore \begin {align*} \frac {1}{\sqrt {g}} \, dy &= \left (\sqrt {f}\right )\,dx\\ -\frac {1}{\sqrt {g}} \, dy &= \left (\sqrt {f}\right )\,dx \end {align*}
Replacing \(f(x),g(y)\) by their values gives \begin {align*} \frac {1}{\sqrt {y^{3}}} \, dy &= \left (\sqrt {\frac {1}{x}}\right )\,dx\\ -\frac {1}{\sqrt {y^{3}}} \, dy &= \left (\sqrt {\frac {1}{x}}\right )\,dx \end {align*}
Integrating now gives the solutions. \begin {align*} \int \frac {1}{\sqrt {y^{3}}}d y &= \int \sqrt {\frac {1}{x}}d x +c_{1}\\ \int -\frac {1}{\sqrt {y^{3}}}d y &= \int \sqrt {\frac {1}{x}}d x +c_{1} \end {align*}
Integrating gives \begin {align*} -\frac {2 y}{\sqrt {y^{3}}} &= 2 x \sqrt {\frac {1}{x}}+c_{1}\\ \frac {2 y}{\sqrt {y^{3}}} &= 2 x \sqrt {\frac {1}{x}}+c_{1} \end {align*}
Therefore \begin{align*} y &= \frac {8 x \sqrt {\frac {1}{x}}+4 c_{1}}{8 x^{3} \left (\frac {1}{x}\right )^{\frac {3}{2}}+6 x \sqrt {\frac {1}{x}}\, c_{1}^{2}+c_{1}^{3}+12 c_{1} x} \\ y &= \frac {8 x \sqrt {\frac {1}{x}}+4 c_{1}}{8 x^{3} \left (\frac {1}{x}\right )^{\frac {3}{2}}+6 x \sqrt {\frac {1}{x}}\, c_{1}^{2}+c_{1}^{3}+12 c_{1} x} \\ \end{align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {8 x \sqrt {\frac {1}{x}}+4 c_{1}}{8 x^{3} \left (\frac {1}{x}\right )^{\frac {3}{2}}+6 x \sqrt {\frac {1}{x}}\, c_{1}^{2}+c_{1}^{3}+12 c_{1} x} \\ \tag{2} y &= \frac {8 x \sqrt {\frac {1}{x}}+4 c_{1}}{8 x^{3} \left (\frac {1}{x}\right )^{\frac {3}{2}}+6 x \sqrt {\frac {1}{x}}\, c_{1}^{2}+c_{1}^{3}+12 c_{1} x} \\ \end{align*}
Verification of solutions
\[ y = \frac {8 x \sqrt {\frac {1}{x}}+4 c_{1}}{8 x^{3} \left (\frac {1}{x}\right )^{\frac {3}{2}}+6 x \sqrt {\frac {1}{x}}\, c_{1}^{2}+c_{1}^{3}+12 c_{1} x} \] Verified OK.
\[ y = \frac {8 x \sqrt {\frac {1}{x}}+4 c_{1}}{8 x^{3} \left (\frac {1}{x}\right )^{\frac {3}{2}}+6 x \sqrt {\frac {1}{x}}\, c_{1}^{2}+c_{1}^{3}+12 c_{1} x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{3}-x {y^{\prime }}^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\sqrt {y x}\, y}{x}, y^{\prime }=-\frac {\sqrt {y x}\, y}{x}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\sqrt {y x}\, y}{x} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\sqrt {y x}\, y}{x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE <- 1st_order WeierstrassP successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 27
dsolve(diff(y(x),x)^2=y(x)^3/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {WeierstrassP}\left (1, 0, 0\right ) 2^{\frac {2}{3}}}{\left (\sqrt {x}\, 2^{\frac {1}{3}}+c_{1} \right )^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.073 (sec). Leaf size: 42
DSolve[(y'[x])^2==y[x]^3/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {4}{\left (-2 \sqrt {x}+c_1\right ){}^2} \\ y(x)\to \frac {4}{\left (2 \sqrt {x}+c_1\right ){}^2} \\ y(x)\to 0 \\ \end{align*}