6.9.22 problem 22

Internal problem ID [1778]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 05:19:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 x +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: 0.003 (sec). Leaf size: 14

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

\[ y = {\mathrm e}^{x} \left (c_2 \,x^{3}+c_1 \right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 23

ode=x*D[y[x],{x,2}]-(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 {1}{3} e^x \left (c_2 x^3+3 c_1\right ) \end{align*}

✗ Sympy

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

False