Internal problem ID [8084]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 504.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {x y^{2} \left (y^{\prime }\right )^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.389 (sec). Leaf size: 544
dsolve(x*y(x)^2*diff(y(x),x)^2-(y(x)^3+x^3-a)*diff(y(x),x)+x^2*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \int _{\textit {\_b}}^{x}\frac {y \relax (x )^{3}-a +\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-2 y \relax (x )^{3} \textit {\_a}^{3}+y \relax (x )^{6}-2 \textit {\_a}^{3} a -2 a y \relax (x )^{3}+a^{2}}}{2 \sqrt {\textit {\_a}^{6}-2 y \relax (x )^{3} \textit {\_a}^{3}+y \relax (x )^{6}-2 \textit {\_a}^{3} a -2 a y \relax (x )^{3}+a^{2}}\, \textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}\frac {\textit {\_f}^{2} \left (3 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{3}+\textit {\_f}^{3}+a \right )}{\left (\textit {\_a}^{6}-2 \textit {\_a}^{3} \textit {\_f}^{3}+\textit {\_f}^{6}-2 \textit {\_a}^{3} a -2 \textit {\_f}^{3} a +a^{2}\right )^{\frac {3}{2}}}d \textit {\_a} \right ) \sqrt {\textit {\_f}^{6}-2 \textit {\_f}^{3} x^{3}+x^{6}-2 \textit {\_f}^{3} a -2 a \,x^{3}+a^{2}}-1\right )}{\sqrt {\textit {\_f}^{6}-2 \textit {\_f}^{3} x^{3}+x^{6}-2 \textit {\_f}^{3} a -2 a \,x^{3}+a^{2}}}d \textit {\_f} +c_{1} = 0 \\ \int _{\textit {\_b}}^{x}-\frac {-y \relax (x )^{3}-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-2 y \relax (x )^{3} \textit {\_a}^{3}+y \relax (x )^{6}-2 \textit {\_a}^{3} a -2 a y \relax (x )^{3}+a^{2}}+a}{2 \sqrt {\textit {\_a}^{6}-2 y \relax (x )^{3} \textit {\_a}^{3}+y \relax (x )^{6}-2 \textit {\_a}^{3} a -2 a y \relax (x )^{3}+a^{2}}\, \textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}\frac {\textit {\_f}^{2} \left (3 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{3}+\textit {\_f}^{3}+a \right )}{\left (\textit {\_a}^{6}-2 \textit {\_a}^{3} \textit {\_f}^{3}+\textit {\_f}^{6}-2 \textit {\_a}^{3} a -2 \textit {\_f}^{3} a +a^{2}\right )^{\frac {3}{2}}}d \textit {\_a} \right ) \sqrt {\textit {\_f}^{6}-2 \textit {\_f}^{3} x^{3}+x^{6}-2 \textit {\_f}^{3} a -2 a \,x^{3}+a^{2}}-1\right )}{\sqrt {\textit {\_f}^{6}-2 \textit {\_f}^{3} x^{3}+x^{6}-2 \textit {\_f}^{3} a -2 a \,x^{3}+a^{2}}}d \textit {\_f} +c_{1} = 0 \\ y \relax (x ) = \left (x^{\frac {3}{2}} c_{1}+x^{3}+a \right )^{\frac {1}{3}} \\ y \relax (x ) = -\frac {\left (x^{\frac {3}{2}} c_{1}+x^{3}+a \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (x^{\frac {3}{2}} c_{1}+x^{3}+a \right )^{\frac {1}{3}}}{2} \\ y \relax (x ) = -\frac {\left (x^{\frac {3}{2}} c_{1}+x^{3}+a \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (x^{\frac {3}{2}} c_{1}+x^{3}+a \right )^{\frac {1}{3}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.41 (sec). Leaf size: 194
DSolve[x^2*y[x] - (-a + x^3 + y[x]^3)*y'[x] + x*y[x]^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{a+(-1+c_1) x^3}}{\sqrt [3]{1-\frac {1}{c_1}}} \\ y(x)\to 0 \\ y(x)\to \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2} \\ y(x)\to (-1)^{2/3} \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2} \\ y(x)\to \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \\ y(x)\to (-1)^{2/3} \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \\ \end{align*}