74.12.44 problem 44

Internal problem ID [16272]
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 : 44
Date solved : Thursday, March 13, 2025 at 08:09:07 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.087 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 0.774 (sec). Leaf size: 138

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

\[ y(t)\to \cos (2 t) \left (-\int _1^0-\frac {\cos (K[1]) \sin (K[1]) (\cos (K[1])+\sin (K[1]))}{\cos (K[1])-\sin (K[1])}dK[1]\right )+\cos (2 t) \int _1^t-\frac {\cos (K[1]) \sin (K[1]) (\cos (K[1])+\sin (K[1]))}{\cos (K[1])-\sin (K[1])}dK[1]-\sin (2 t) \int _1^0\frac {1}{2} (\cos (K[2])+\sin (K[2]))^2dK[2]+\sin (2 t) \int _1^t\frac {1}{2} (\cos (K[2])+\sin (K[2]))^2dK[2]+\cos (2 t) \]

✓ Sympy. Time used: 2.467 (sec). Leaf size: 54

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

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