Internal problem ID [3725]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1001.
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 (a -x^{3}-y^{3}\right ) y^{\prime }+x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.197 (sec). Leaf size: 190
dsolve(x*y(x)^2*diff(y(x),x)^2+(a-x^3-y(x)^3)*diff(y(x),x)+x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \int _{\textit {\_b}}^{y \relax (x )}\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 \relax (x )}{2}-c_{1} = 0 \\ \int _{\textit {\_b}}^{y \relax (x )}\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 \relax (x )}{2}-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: 0.393 (sec). Leaf size: 194
DSolve[x y[x]^2 (y'[x])^2 +(a-x^3-y[x]^3) y'[x]+x^2 y[x]==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*}