89.20.13 problem 13

Internal problem ID [24769]
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. Oral Exercises at page 154
Problem number : 13
Date solved : Thursday, October 02, 2025 at 10:47:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to \left (-\frac {x}{4}+c_1\right ) \cos (2 x)+\frac {1}{8} (1+16 c_2) \sin (x) \cos (x) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 19

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

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