56.1.35 problem 36

Internal problem ID [8747]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 36
Date solved : Friday, March 14, 2025 at 01:49:25 AM
CAS classification : [_Clairaut]

\begin{align*} x f^{\prime }-f&=\frac {{f^{\prime }}^{2} \left (1-{f^{\prime }}^{\lambda }\right )^{2}}{\lambda ^{2}} \end{align*}

✓ Maple. Time used: 0.432 (sec). Leaf size: 318

ode:=x*diff(f(x),x)-f(x) = diff(f(x),x)^2/lambda^2*(1-diff(f(x),x)^lambda)^2; 
dsolve(ode,f(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 15.191 (sec). Leaf size: 30

ode=x*D[ f[x],x]-f[x]==D[ f[x],x]^2/\[Lambda]^2*(1-D[ f[x],x]^\[Lambda])^2; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
cg = symbols("cg") 
f = Function("f") 
ode = Eq(x*Derivative(f(x), x) - f(x) - (1 - Derivative(f(x), x)**cg)**2*Derivative(f(x), x)**2/cg**2,0) 
ics = {} 
dsolve(ode,func=f(x),ics=ics)
 

NotImplementedError : multiple generators [_X0, _X0**cg] 
No algorithms are implemented to solve equation -_X0**2*_X0**(2*cg) + 2*_X0**2*_X0**cg - _X0**2 + _X0*cg**2*x - cg**2*f(x)