89.21.14 problem 16

Internal problem ID [24786]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 154
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:48:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\sin \left (3 x \right )+4 \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)+y(x) = sin(3*x)+4*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\sin \left (3 x \right )}{8}+\cos \left (x \right ) \left (c_1 +1\right )+\left (2 x +c_2 \right ) \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+y[x]== Sin[3*x]+4*Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (2+c_1) \cos (x)+\frac {1}{8} \sin (x) (16 x-2 \cos (2 x)-3+8 c_2) \end{align*}

✓ Sympy. Time used: 0.060 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - sin(3*x) - 4*cos(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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