Internal
problem
ID
[8883]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
25
Date
solved
:
Wednesday, March 05, 2025 at 07:07:22 AM
CAS
classification
:
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Solve
Time used: 0.487 (sec)
This is second order ode with missing dependent variable
Then
Hence the ode becomes
Which is now solved for
The ode
is separable as it can be written as
Where
Integrating gives
In summary, these are the solution found for
For solution
Since the ode has the form
In summary, these are the solution found for
Will add steps showing solving for IC soon.
Summary of solutions found
ode:=(x^2+1)*diff(diff(y(x),x),x)+diff(y(x),x)^2 = 0; dsolve(ode,y(x), singsol=all);
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)^2/(_a^2+1), _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
Maple step by step
ode=(1+x^2)*D[y[x],{x,2}]+(D[y[x],x])^2==0; ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
from sympy import * x = symbols("x") y = Function("y") ode = Eq((x**2 + 1)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2,0) ics = {} dsolve(ode,func=y(x),ics=ics)