Internal problem ID [8839]
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]
\[ \boxed {x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y=0} \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 307
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 \left (x \right ) = \left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} y \left (x \right ) = \left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} y \left (x \right ) = -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2} y \left (x \right ) = -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2} y \left (x \right ) = -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2} y \left (x \right ) = -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}}}{2} y \left (x \right ) = 0 \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} +\frac {\ln \left (x \right )}{2}-c_{1} = 0 \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} -\frac {\ln \left (x \right )}{2}-c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 0.488 (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*}