Internal problem ID [7079]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 36.
ODE order: 1.
ODE degree: 0.
CAS Maple gives this as type [_Clairaut]
\[ \boxed {x f^{\prime }-f-\frac {{f^{\prime }}^{2} \left (1-{f^{\prime }}^{\lambda }\right )^{2}}{\lambda ^{2}}=0} \]
✓ Solution by Maple
Time used: 0.468 (sec). Leaf size: 318
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 \left (x \right ) &= 0 \\ f \left (x \right ) &= \frac {\lambda ^{2} x^{2} \left (2 \lambda \,{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }+{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }-1\right )}{4 \left (\lambda \,{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }+{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }-1\right )^{2} \left ({\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }-1\right )} \\ f \left (x \right ) &= c_{1} x -\frac {c_{1}^{2} \left (-1+c_{1}^{\lambda }\right )^{2}}{\lambda ^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 15.811 (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*}