76.17.6 problem 15

Internal problem ID [17621]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 15
Date solved : Monday, March 31, 2025 at 04:22:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=3 \sec \left (2 t \right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)+4*y(t) = 3*sec(2*t)^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {3 \ln \left (\sec \left (2 t \right )+\tan \left (2 t \right )\right ) \sin \left (2 t \right )}{4}-\frac {3}{4} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 36

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

\[ y(t)\to c_1 \cos (2 t)+\frac {3}{4} \sin (2 t) \coth ^{-1}(\sin (2 t))+c_2 \sin (2 t)-\frac {3}{4} \]

✓ Sympy. Time used: 0.409 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)) - 3/cos(2*t)**2,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \cos {\left (2 t \right )} + \left (C_{1} - \frac {3 \log {\left (\sin {\left (2 t \right )} - 1 \right )}}{8} + \frac {3 \log {\left (\sin {\left (2 t \right )} + 1 \right )}}{8}\right ) \sin {\left (2 t \right )} - \frac {3}{4} \]