74.12.44 problem 44

Internal problem ID [16351]
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 : Tuesday, January 28, 2025 at 09:06:27 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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