32.8.18 problem Exercise 21.22, page 231

Internal problem ID [5967]
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 : Sunday, March 30, 2025 at 10:28:28 AM
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: 0.317 (sec). Leaf size: 36

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

\[ y(x)\to \frac {1}{4} \left (2 e^{-x}+\sin (x)-2 x \cos (x)+4 c_1 \cos (x)+4 c_2 \sin (x)\right ) \]

✓ Sympy. Time used: 0.101 (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)
 

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