67.9.29 problem 14.3 (e)

Internal problem ID [16577]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.3 (e)
Date solved : Thursday, October 02, 2025 at 01:36:34 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y&=8 \,{\mathrm e}^{2 x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=x*diff(diff(y(x),x),x)+(2*x+2)*diff(y(x),x)+2*y(x) = 8*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 31

ode=x*D[y[x],{x,2}]+(2+2*x)*D[y[x],x]+2*y[x]==8*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (2*x + 2)*Derivative(y(x), x) + 2*y(x) - 8*exp(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*Derivative(y(x), (x, 2))/2 - y(x) + 4*exp(2*x))/(x + 1) cannot be solved by the factorable group method