Internal problem ID [8050]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 470.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}+x^{3} y^{\prime }-x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.476 (sec). Leaf size: 91
dsolve(y(x)*diff(y(x),x)^2+x^3*diff(y(x),x)-x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {i x^{2}}{2} \\ y \relax (x ) = \frac {i x^{2}}{2} \\ y \relax (x ) = 0 \\ y \relax (x ) = -\frac {\sqrt {-4 c_{1} x^{2}+c_{1}^{2}}}{4} \\ y \relax (x ) = \frac {\sqrt {-4 c_{1} x^{2}+c_{1}^{2}}}{4} \\ y \relax (x ) = -\frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ y \relax (x ) = \frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.721 (sec). Leaf size: 169
DSolve[-(x^2*y[x]) + x^3*y'[x] + y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \tanh ^{-1}\left (\frac {\sqrt {x^4+4 y(x)^2}}{x^2}\right )}{2 x \sqrt {x^4+4 y(x)^2}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \tanh ^{-1}\left (\frac {\sqrt {x^4+4 y(x)^2}}{x^2}\right )}{2 x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to -\frac {i x^2}{2} \\ y(x)\to \frac {i x^2}{2} \\ \end{align*}