67.4.26 problem Problem 4(b)

Internal problem ID [14070]
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(b)
Date solved : Tuesday, January 28, 2025 at 06:13:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 0 & 0\le t <1 \\ 1 & 1\le t <2 \\ -1 & 2\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 14.206 (sec). Leaf size: 121

dsolve([diff(y(t),t$2)-3*diff(y(t),t)+2*y(t)=piecewise(0<=t and t<1,0,t>=1 and t<2,1,t>=2,-1 ),y(0) = 3, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = \left \{\begin {array}{cc} 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t} & t <1 \\ 7 \,{\mathrm e}-4 \,{\mathrm e}^{2}+\frac {1}{2} & t =1 \\ 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t}-{\mathrm e}^{t -1}+\frac {{\mathrm e}^{2 t -2}}{2}+\frac {1}{2} & t <2 \\ \frac {15 \,{\mathrm e}^{2}}{2}-4 \,{\mathrm e}^{4}-{\mathrm e}-\frac {1}{2} & t =2 \\ 7 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{2 t}+2 \,{\mathrm e}^{t -2}-{\mathrm e}^{t -1}-{\mathrm e}^{2 t -4}+\frac {{\mathrm e}^{2 t -2}}{2}-\frac {1}{2} & 2<t \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 109

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^t \left (7-4 e^t\right ) & t\leq 1 \\ \frac {1}{2} \left (1-2 e^{t-1}+14 e^t-8 e^{2 t}+e^{2 t-2}\right ) & 1<t\leq 2 \\ \frac {1}{2} \left (-1+4 e^{t-2}-2 e^{t-1}+14 e^t-8 e^{2 t}-2 e^{2 t-4}+e^{2 t-2}\right ) & \text {True} \\ \end {array} \\ \end {array} \]