87.14.26 problem 26

Internal problem ID [23545]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 26
Date solved : Thursday, October 02, 2025 at 09:42:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=-x \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

ode:=(x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 x +c_2 \left (x^{2}-1\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 21

ode=(x^2+1)*D[y[x],{x,2}]-2*x*D[y[x],x]+2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 x-c_1 (x-i)^2 \end{align*}

✗ Sympy

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

False