87.16.2 problem 2

Internal problem ID [23571]
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 119
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:43:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&={\mathrm e}^{x}+{\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = exp(x)+exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (2 \,{\mathrm e}^{3 x}+18 c_1 x +9 x^{2}+18 c_2 \right ) {\mathrm e}^{-2 x}}{18} \]

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 33

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==Exp[x]+Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^x}{9}+e^{-2 x} \left (\frac {x^2}{2}+c_2 x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - exp(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x}{2}\right )\right ) e^{- 2 x} + \frac {e^{x}}{9} \]