Internal problem ID [4232]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1000.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }=-a^{2} x} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 153
dsolve(x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)+a^2*x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \sqrt {-2 a x} y \left (x \right ) = -\sqrt {-2 a x} y \left (x \right ) = \sqrt {2}\, \sqrt {a x} y \left (x \right ) = -\sqrt {2}\, \sqrt {a x} y \left (x \right ) = {\mathrm e}^{\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (16 a^{2} x \,{\mathrm e}^{2 c_{1}} {\mathrm e}^{2 \textit {\_Z}}+x^{3} {\mathrm e}^{2 \textit {\_Z}}-4 \,{\mathrm e}^{3 \textit {\_Z}} {\mathrm e}^{2 c_{1}}\right )}{2}-\frac {\ln \left (x \right )}{2}} y \left (x \right ) = {\mathrm e}^{-\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (x^{2} \left (16 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 \textit {\_Z}} x^{2} a^{2}+{\mathrm e}^{2 \textit {\_Z}}-4 \,{\mathrm e}^{3 \textit {\_Z}} {\mathrm e}^{-2 c_{1}} x \right )\right )}{2}+\frac {\ln \left (x \right )}{2}} \end{align*}
✓ Solution by Mathematica
Time used: 22.383 (sec). Leaf size: 219
DSolve[x y[x]^2 (y'[x])^2 - y[x]^3 y'[x]+a^2 x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-2 a^2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} y(x)\to \sqrt {-2 a^2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} y(x)\to -\frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} y(x)\to \frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} y(x)\to -\sqrt {2} \sqrt {a} \sqrt {x} y(x)\to -i \sqrt {2} \sqrt {a} \sqrt {x} y(x)\to i \sqrt {2} \sqrt {a} \sqrt {x} y(x)\to \sqrt {2} \sqrt {a} \sqrt {x} \end{align*}