46.7.10 problem 27

Internal problem ID [7371]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 27
Date solved : Monday, January 27, 2025 at 02:51:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 8 t^{2} & 0<t <5 \\ 0 & 5<t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (1\right )&=1+\cos \left (2\right )\\ y^{\prime }\left (1\right )&=4-2 \sin \left (2\right ) \end{align*}

✓ Solution by Maple

Time used: 0.767 (sec). Leaf size: 40

dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<t and t<5,8*t^2,t>5,0),y(1) = 1+cos(2), D(y)(1) = 4-2*sin(2)],y(t), singsol=all)
 

\[ y = \cos \left (2 t \right )+\left (\left \{\begin {array}{cc} 2 t^{2}-1 & t <5 \\ 10 \sin \left (-10+2 t \right )+49 \cos \left (-10+2 t \right ) & 5\le t \end {array}\right .\right ) \]

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 51

DSolve[{D[y[t],{t,2}]+4*y[t]==Piecewise[{{8*t^2,0<t<5},{0,t>5}}],{y[1]==1+Cos[2],Derivative[1][y][1]==4-2*Sin[2]}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 2 t^2+\cos (2 t)-1 & 0<t\leq 5 \\ 49 \cos (2 (t-5))+\cos (2 t)-10 \sin (10-2 t) & t>5 \\ \end {array} \\ \end {array} \]