89.16.17 problem 17

Internal problem ID [24667]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 17
Date solved : Thursday, October 02, 2025 at 10:46:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&={\mathrm e}^{-x} \left (2 \sin \left (x \right )+4 \cos \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.068 (sec). Leaf size: 17

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

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