76.22.1 problem 14

Internal problem ID [17782]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 11:02:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 15.622 (sec). Leaf size: 32

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

\[ y = \frac {-\int _{0}^{t}\sin \left (w \left (-t +\textit {\_U1} \right )\right ) g \left (\textit {\_U1} \right )d \textit {\_U1} +\sin \left (w t \right )}{w} \]

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 114

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

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