Internal problem ID [9215]
Internal file name [OUTPUT/8151_Monday_June_06_2022_01_57_28_AM_14475903/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 882.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
\[ \boxed {y^{\prime }+\frac {\left (-108 x^{\frac {3}{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{216}=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=y^{3} \sqrt {x}+\frac {\left (-108 x^{3}+216\right ) \sqrt {x}\, y^{2}}{216}+\frac {\left (18 x^{6}-72 x^{3}\right ) \sqrt {x}\, y}{216}+\frac {\left (-x^{9}+6 x^{6}+108 x^{\frac {3}{2}}+216\right ) \sqrt {x}}{216}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -\frac {x^{\frac {19}{2}}}{216}+\frac {x^{\frac {13}{2}}}{36}+\frac {x^{2}}{2}+\sqrt {x}\\ f_1(x) &= \frac {x^{\frac {13}{2}}}{12}-\frac {x^{\frac {7}{2}}}{3}\\ f_2(x) &= -\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}\\ f_3(x) &= \sqrt {x} \end {align*}
Since \(f_2(x)=-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}}{3 \sqrt {x}} \right ) \\ &= u \left (x \right )+\frac {x^{3}}{6}-\frac {1}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \sqrt {x}\, u \left (x \right )^{3}-\frac {\sqrt {x}\, u \left (x \right )}{3}+\frac {29 \sqrt {x}}{27}\tag {2} \end {align*}
The above ODE (2) can now be solved as separable.
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \sqrt {x}\, \left (u^{3}-\frac {1}{3} u +\frac {29}{27}\right ) \end {align*}
Where \(f(x)=\sqrt {x}\) and \(g(u)=u^{3}-\frac {1}{3} u +\frac {29}{27}\). Integrating both sides gives \begin{align*} \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du &= \sqrt {x} \,d x \\ \int { \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du} &= \int {\sqrt {x} \,d x} \\ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}&=\frac {2 x^{\frac {3}{2}}}{3}+c_{2} \\ \end{align*} Which results in \[ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}=\frac {2 x^{\frac {3}{2}}}{3}+c_{2} \] The solution is \[ \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {2 x^{\frac {3}{2}}}{3}-c_{2} = 0 \] Substituting \(u=y-\frac {-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}}{3 \sqrt {x}}\) in the above solution gives \begin {align*} \int _{}^{y-\frac {-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}}{3 \sqrt {x}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {2 x^{\frac {3}{2}}}{3}-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y-\frac {-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}}{3 \sqrt {x}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {2 x^{\frac {3}{2}}}{3}-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \int _{}^{y-\frac {-\frac {x^{\frac {7}{2}}}{2}+\sqrt {x}}{3 \sqrt {x}}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\frac {2 x^{\frac {3}{2}}}{3}-c_{2} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {\left (-108 x^{\frac {3}{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{216}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\left (-108 x^{\frac {3}{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{216} \end {array} \]
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 Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
dsolve(diff(y(x),x) = -1/216*(-108*x^(3/2)-216-216*y(x)^2+72*x^3*y(x)-6*x^6-216*y(x)^3+108*x^3*y(x)^2-18*y(x)*x^6+x^9)*x^(1/2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{3}}{6}-\frac {1}{3}+\frac {29 \operatorname {RootOf}\left (2 x^{\frac {3}{2}}-243 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+9 c_{1} \right )}{9} \]
✓ Solution by Mathematica
Time used: 0.2 (sec). Leaf size: 119
DSolve[y'[x] == -1/216*(Sqrt[x]*(-216 - 108*x^(3/2) - 6*x^6 + x^9 + 72*x^3*y[x] - 18*x^6*y[x] - 216*y[x]^2 + 108*x^3*y[x]^2 - 216*y[x]^3)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {1}{2} \left (2 \sqrt {x}-x^{7/2}\right )+3 \sqrt {x} y(x)}{\sqrt [3]{29} \sqrt [3]{x^{3/2}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {2}{27} 29^{2/3} \sqrt {x} \left (x^{3/2}\right )^{2/3}+c_1,y(x)\right ] \]