9.1.5 problem problem 42

Internal problem ID [932]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.2, Higher-Order Linear Differential Equations. General solutions of Linear Equations. Page 288
Problem number : problem 42
Date solved : Thursday, March 13, 2025 at 03:53:33 PM
CAS classification : [_Gegenbauer]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 14

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);
 

\[ y = c_2 \,x^{2}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 39

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

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

✗ Sympy

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

False