81.11.14 problem 15-13

Internal problem ID [21638]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 15. Method of undetermined coefficients. Page 337.
Problem number : 15-13
Date solved : Thursday, October 02, 2025 at 07:59:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

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

\[ y = \frac {\left (-x +4 c_1 \right ) \cos \left (2 x \right )}{4}+\frac {\left (16 c_2 +1\right ) \sin \left (2 x \right )}{16}+\frac {\sin \left (x \right )}{3} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \frac {\sin (x)}{3}-\frac {1}{5} \sin (3 x)+c_1 \cos (2 x)+c_2 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 24

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

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