87.14.5 problem 5

Internal problem ID [23524]
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 : 5
Date solved : Thursday, October 02, 2025 at 09:42:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {{\mathrm e}^{-x}}{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

\[ y = \frac {c_1 \,{\mathrm e}^{-x}+c_2 \left (x^{2}-2 x +2\right )}{x} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to \frac {e^{-x} \left (c_2 e^x \left (x^2-2 x+2\right )+c_1\right )}{x} \end{align*}

✗ Sympy

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

False