56.1.35 problem 36
Internal
problem
ID
[8747]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
36
Date
solved
:
Monday, January 27, 2025 at 04:46:16 PM
CAS
classification
:
[_Clairaut]
\begin{align*} x f^{\prime }-f&=\frac {{f^{\prime }}^{2} \left (1-{f^{\prime }}^{\lambda }\right )^{2}}{\lambda ^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.432 (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 &= 0 \\
f &= \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 &= c_{1} x -\frac {c_{1}^{2} \left (-1+c_{1}^{\lambda }\right )^{2}}{\lambda ^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 15.191 (sec). Leaf size: 30
DSolve[x*D[ f[x],x]-f[x]==D[ f[x],x]^2/\[Lambda]^2*(1-D[ f[x],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*}