89.16.26 problem 26

Internal problem ID [24676]
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. Exercises at page 140
Problem number : 26
Date solved : Thursday, October 02, 2025 at 10:46:57 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 34

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

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

✓ Sympy. Time used: 0.421 (sec). Leaf size: 22

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

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