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]
Solve
Time used: 0.149 (sec)
This is Clairaut ODE. It has the form
Where \(g\) is function of \(f'(x)\). Let \(p=f^{\prime }\) the ode becomes
Solving for \(f\) from the above results in
The above ode is a Clairaut ode which is now solved.
We start by replacing \(f^{\prime }\) by \(p\) which gives
Writing the ode as
We now write \(g\equiv g\left ( p\right ) \) to make notation simpler but we should always remember that \(g\) is function of \(p\) which in turn is function of \(x\). Hence the above becomes
Then we see that
Taking derivative of (1) w.r.t. \(x\) gives
Where \(g'\) is derivative of \(g\left ( p\right ) \) w.r.t. \(p\).
The general solution is given by
Substituting this in (1) gives the general solution as
The singular solution is found from solving for \(p\) from
And substituting the result back in (1). Since we found above that \(g=-\frac {p^{2} \left (p^{2 \lambda }-2 p^{\lambda }+1\right )}{\lambda ^{2}}\), then the above equation becomes
Unable to solve for \(p\). No singular solutions can be found.
Summary of solutions found
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);
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying dAlembert <- 1st order, parametric methods successful <- dAlembert successful`
Maple step by step
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]
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)