26.8.18 problem Exercise 21.22, page 231

Internal problem ID [7102]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 21. Undetermined Coefficients
Problem number : Exercise 21.22, page 231
Date solved : Tuesday, September 30, 2025 at 04:21:28 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to \cos (x) \int _1^x\left (-\sin (K[1])-e^{-K[1]}\right ) \sin (K[1])dK[1]+\sin (x) \int _1^x\cos (K[2]) \left (\sin (K[2])+e^{-K[2]}\right )dK[2]+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 22

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

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