89.26.10 problem 12

Internal problem ID [24877]
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. Miscellaneous Exercises at page 177
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:48:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y-3 y^{\prime }+y^{\prime \prime }&=\sin \left ({\mathrm e}^{-x}\right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-3*diff(y(x),x)+2*y(x) = sin(exp(-x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 29

ode=D[y[x],{x,2}]-3*D[y[x],x]+2*y[x]== Sin[Exp[-x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.797 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {\cos {\left (e^{- x} \right )}}{4}\right ) e^{x} + \left (C_{2} - \frac {\operatorname {Ci}{\left (e^{- x} \right )}}{4}\right ) e^{- x} + \frac {\sin {\left (e^{- x} \right )}}{4} \]