74.12.41 problem 41

Internal problem ID [16269]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 41
Date solved : Thursday, March 13, 2025 at 08:08:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)+4*y(t) = sec(2*t)^2; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 35

ode=D[y[t],{t,2}]+4*y[t]==Sec[2*t]^2; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{4} \sin (t) \left (-2 \sin (t)+(4-i \pi ) \cos (t)+2 \cos (t) \coth ^{-1}(\sin (2 t))\right ) \]

✓ Sympy. Time used: 0.426 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)) - 1/cos(2*t)**2,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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