Internal problem ID [6326]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 36.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [_Clairaut]
Solve \begin {gather*} \boxed {f^{\prime } x -f-\frac {\left (f^{\prime }\right )^{2} \left (1-\left (f^{\prime }\right )^{\lambda }\right )^{2}}{\lambda ^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.282 (sec). Leaf size: 193
dsolve(x*diff(f(x),x)-f(x)=diff(f(x),x)^2/lambda^2*(1-diff(f(x),x)^lambda)^2,f(x), singsol=all)
\begin{align*} f \relax (x ) = 0 \\ f \relax (x ) = -\frac {\RootOf \left (\textit {\_Z} \left (2 \lambda \,\textit {\_Z}^{\lambda +1}-2 \lambda \,\textit {\_Z}^{2 \lambda +1}+x \,\lambda ^{2}+4 \textit {\_Z}^{\lambda +1}-2 \textit {\_Z}^{2 \lambda +1}-2 \textit {\_Z} \right )\right )^{\lambda +2}}{\lambda }+\frac {\RootOf \left (\textit {\_Z} \left (2 \lambda \,\textit {\_Z}^{\lambda +1}-2 \lambda \,\textit {\_Z}^{2 \lambda +1}+x \,\lambda ^{2}+4 \textit {\_Z}^{\lambda +1}-2 \textit {\_Z}^{2 \lambda +1}-2 \textit {\_Z} \right )\right )^{2 \lambda +2}}{\lambda }+\frac {\RootOf \left (\textit {\_Z} \left (2 \lambda \,\textit {\_Z}^{\lambda +1}-2 \lambda \,\textit {\_Z}^{2 \lambda +1}+x \,\lambda ^{2}+4 \textit {\_Z}^{\lambda +1}-2 \textit {\_Z}^{2 \lambda +1}-2 \textit {\_Z} \right )\right ) x}{2} \\ f \relax (x ) = c_{1} x -\frac {c_{1}^{2} \left (1-c_{1}^{\lambda }\right )^{2}}{\lambda ^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.243 (sec). Leaf size: 30
DSolve[x*f'[x]-f[x]==f'[x]^2/\[Lambda]^2*(1-f'[x]^\[Lambda])^2,f[x],x,IncludeSingularSolutions -> True]
\begin{align*} f(x)\to c_1 \left (x-\frac {c_1 \left (-1+c_1{}^{\lambda }\right ){}^2}{\lambda ^2}\right ) \\ f(x)\to 0 \\ \end{align*}