74.12.43 problem 43

Internal problem ID [16350]
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 : 43
Date solved : Tuesday, January 28, 2025 at 09:06:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\tan \left (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: 0.017 (sec). Leaf size: 21

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

\[ y = \sin \left (t \right )+2 \cos \left (t \right )-2+\sin \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 59

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

\[ y(t)\to -\cos (t) \int _1^0-\sin (K[1]) \tan ^2(K[1])dK[1]+\cos (t) \int _1^t-\sin (K[1]) \tan ^2(K[1])dK[1]+\sin (t) (\text {arctanh}(\sin (t))-\sin (t)+1) \]