74.12.41 problem 41

Internal problem ID [16348]
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 : Tuesday, January 28, 2025 at 09:06:11 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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