74.9.9 problem 17

Internal problem ID [16185]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 08:56:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 13

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

\[ y = \sin \left (t \right )+\cos \left (t \right )+t \sin \left (t \right ) \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 47

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

\[ y(t)\to -\sin (t) \int _1^02 \cos ^2(K[1])dK[1]+\sin (t) \int _1^t2 \cos ^2(K[1])dK[1]+\sin (t)+\cos ^3(t) \]