89.23.13 problem 13

Internal problem ID [24817]
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. Miscellaneous Exercises at page 162
Problem number : 13
Date solved : Thursday, October 02, 2025 at 10:48:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y+y^{\prime \prime }&=12 \sin \left (x \right )+12 \sin \left (2 x \right ) \end{align*}

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

ode:=diff(diff(y(x),x),x)+4*y(x) = 12*sin(x)+12*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 -3 x \right ) \cos \left (2 x \right )+\frac {\left (4 c_2 +3\right ) \sin \left (2 x \right )}{4}+4 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 28

ode=D[y[x],{x,2}]+4*y[x]==12*(Sin[x]+Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.065 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 12*sin(x) - 12*sin(2*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} \sin {\left (2 x \right )} + \left (C_{1} - 3 x\right ) \cos {\left (2 x \right )} + 4 \sin {\left (x \right )} \]