89.18.5 problem 5

Internal problem ID [24705]
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. Oral Exercises at page 146
Problem number : 5
Date solved : Thursday, October 02, 2025 at 10:47:17 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to -x+\frac {e^x}{2}-\frac {1}{8} \sin (3 x)+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.055 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x + y(x) - exp(x) - sin(3*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )} - x + \frac {e^{x}}{2} - \frac {\sin {\left (3 x \right )}}{8} \]