14.17.2 problem 20

Internal problem ID [2680]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 20
Date solved : Monday, January 27, 2025 at 06:06:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.783 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 25

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

\[ y(t)\to \frac {1}{4} \left ((t+8) \sin (t)-\left (t^2-4\right ) \cos (t)\right ) \]