73.18.6 problem 27.1 (f)

Internal problem ID [15590]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (f)
Date solved : Tuesday, January 28, 2025 at 08:02:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 8.086 (sec). Leaf size: 20

dsolve([diff(y(t),t$2)+4*y(t)=sin(2*t),y(0) = 3, D(y)(0) = 5],y(t), singsol=all)
 

\[ y = \frac {21 \sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right ) \left (-12+t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 108

DSolve[{D[y[t],{t,2}]+4*y[t]==Sin[2*t],{y[0]==3,Derivative[1][y][0] ==5}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\sin (2 t) \int _1^0\frac {1}{4} \sin (4 K[2])dK[2]+\sin (2 t) \int _1^t\frac {1}{4} \sin (4 K[2])dK[2]+\cos (2 t) \left (-\int _1^0-\frac {1}{2} \sin ^2(2 K[1])dK[1]\right )+\cos (2 t) \int _1^t-\frac {1}{2} \sin ^2(2 K[1])dK[1]+3 \cos (2 t)+5 \sin (t) \cos (t) \]