Internal
problem
ID
[9032]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
48
Date
solved
:
Friday, February 21, 2025 at 09:03:43 PM
CAS
classification
:
[_quadrature]
Solve
Time used: 0.018 (sec)
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
Summary of solutions found
Time used: 0.156 (sec)
Applying change of variables \(y = u \left (x \right ) x\), then the ode becomes
Which is now solved The ode
is separable as it can be written as
Where
Integrating gives
We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is zero, since we had to divide by this above. Solving \(g(u)=0\) or
for \(u \left (x \right )\) gives
Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.
Therefore the solutions found are
Solving for \(u \left (x \right )\) gives
Converting \(u \left (x \right ) = 0\) back to \(y\) gives
Converting \(u \left (x \right ) = \frac {{\mathrm e}^{c_1}}{x}\) back to \(y\) gives
Summary of solutions found
`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 <- differential order: 1; missing y(x) successful`
Solving time : 0.003
(sec)
Leaf size : 5
dsolve(x*diff(y(x),x)^n = 0,y(x),singsol=all)
Solving time : 0.005
(sec)
Leaf size : 15
DSolve[{x*(D[y[x],x])^n==0,{}},y[x],x,IncludeSingularSolutions->True]
Solving time : 0.000
(sec)
Leaf size : 0
Python version: 3.13.1 (main, Dec 4 2024, 18:05:56) [GCC 14.2.1 20240910] Sympy version 1.13.3
from sympy import * x = symbols("x") n = symbols("n") y = Function("y") ode = Eq(x*Derivative(y(x), x)**n,0) ics = {} dsolve(ode,func=y(x),ics=ics)
TypeError : cannot determine truth value of Relational