Internal problem ID [6314]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 24.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y-x y^{\prime }-x^{2} \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.149 (sec). Leaf size: 97
dsolve(y(x)=x*diff(y(x),x)+x^2*diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} \ln \relax (x )-\sqrt {4 y \relax (x )+1}-\frac {\ln \left (-1+\sqrt {4 y \relax (x )+1}\right )}{2}+\frac {\ln \left (1+\sqrt {4 y \relax (x )+1}\right )}{2}-\frac {\ln \left (y \relax (x )\right )}{2}-c_{1} = 0 \\ \ln \relax (x )+\sqrt {4 y \relax (x )+1}+\frac {\ln \left (-1+\sqrt {4 y \relax (x )+1}\right )}{2}-\frac {\ln \left (1+\sqrt {4 y \relax (x )+1}\right )}{2}-\frac {\ln \left (y \relax (x )\right )}{2}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 67
DSolve[y[x]==x*y'[x]+x^2*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \text {ProductLog}\left (-e^{-1-2 c_1} x\right ) \left (2+\text {ProductLog}\left (-e^{-1-2 c_1} x\right )\right ) \\ y(x)\to \frac {1}{4} \text {ProductLog}\left (e^{-1+2 c_1} x\right ) \left (2+\text {ProductLog}\left (e^{-1+2 c_1} x\right )\right ) \\ \end{align*}