39.3.5 problem Problem 12.5

Internal problem ID [6531]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 12. VARIATION OF PARAMETERS. page 104
Problem number : Problem 12.5
Date solved : Wednesday, March 05, 2025 at 12:55:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

\[ x \left (t \right ) = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {1}{8}+\frac {\cos \left (4 t \right )}{24} \]

✓ Mathematica. Time used: 0.123 (sec). Leaf size: 31

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

\[ x(t)\to \frac {1}{24} \cos (4 t)+c_1 \cos (2 t)+c_2 \sin (2 t)+\frac {1}{8} \]

✓ Sympy. Time used: 0.388 (sec). Leaf size: 32

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

\[ x{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )} + \frac {\sin ^{4}{\left (t \right )}}{3} - \frac {\sin ^{2}{\left (t \right )}}{3} + \frac {1}{6} \]