89.20.7 problem 7

Internal problem ID [24763]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Oral Exercises at page 154
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:47:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

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

\[ y = \sin \left (\frac {x}{2}\right ) c_2 +\cos \left (\frac {x}{2}\right ) c_1 +\frac {{\mathrm e}^{-2 x}}{17} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to \frac {e^{-2 x}}{17}+c_1 \cos \left (\frac {x}{2}\right )+c_2 \sin \left (\frac {x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} \sin {\left (\frac {x}{2} \right )} + C_{2} \cos {\left (\frac {x}{2} \right )} + \frac {e^{- 2 x}}{17} \]