88.21.3 problem 3

Internal problem ID [24153]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 149
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:00:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

\[ y = \frac {c_1}{x}+c_2 \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \frac {2 c_2 e^{x+1} x+c_1}{\sqrt {2 e} x} \end{align*}

✗ Sympy

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

False