89.24.15 problem 15

Internal problem ID [24852]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Exercises at page 171
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:48:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

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

\[ y = -{\mathrm e}^{x} \cos \left ({\mathrm e}^{-x}\right )-\sin \left ({\mathrm e}^{-x}\right )+{\mathrm e}^{-x} c_1 +\left (c_2 -1\right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 39

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

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

✓ Sympy. Time used: 2.726 (sec). Leaf size: 26

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

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