67.4.25 problem Problem 4(a)

Internal problem ID [14069]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 4(a)
Date solved : Tuesday, January 28, 2025 at 06:13:34 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }&=\left \{\begin {array}{cc} 4 & 0\le t <1 \\ 6 & 1\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-6\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 13.396 (sec). Leaf size: 52

dsolve([diff(y(t),t$2)-2*diff(y(t),t)=piecewise(0<=t and t<1,4,t>=1,6),y(0) = -6, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = -\frac {\left (\left \{\begin {array}{cc} 15+4 t -3 \,{\mathrm e}^{2 t} & t <1 \\ 20-3 \,{\mathrm e}^{2} & t =1 \\ 14+6 t -3 \,{\mathrm e}^{2 t}-{\mathrm e}^{2 t -2} & 1<t \end {array}\right .\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 68

DSolve[{D[y[t],{t,2}]-2*D[y[t],t]==Piecewise[{{4,0<=t<1},{6,t>=1}}],{y[0]==-6,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} \left (-13+e^{2 t}\right ) & t\leq 0 \\ \frac {1}{2} \left (-4 t+3 e^{2 t}-15\right ) & 0<t\leq 1 \\ \frac {1}{2} \left (-6 t+3 e^{2 t}+e^{2 t-2}-14\right ) & \text {True} \\ \end {array} \\ \end {array} \]