6.10.2 problem 2

Internal problem ID [1806]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:19:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.304 (sec). Leaf size: 27

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

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