Internal problem ID [10986]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y-\left (\left \{\begin {array}{cc} 0 & 0\le t <1 \\ 1 & 1\le t <2 \\ -1 & 2\le t \end {array}\right .\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 74
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)
\[ y \left (t \right ) = -4 \,{\mathrm e}^{2 t}+7 \,{\mathrm e}^{t}-\frac {\left (\left \{\begin {array}{cc} 0 & t <1 \\ -1+2 \,{\mathrm e}^{t -1}-{\mathrm e}^{2 t -2} & t <2 \\ 1+2 \,{\mathrm e}^{t -1}-{\mathrm e}^{2 t -2}-4 \,{\mathrm e}^{t -2}+2 \,{\mathrm e}^{-4+2 t} & 2\le t \end {array}\right .\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 100
DSolve[{y''[t]-3*y'[t]+2*y[t]==Piecewise[{{0,0<=t<1},{1,1<=t<2},{-1,t>=2}}],{y[0]==3,y'[0]==-1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} e^t \left (7-4 e^t\right ) & t\leq 1 \\ \frac {1}{2} \left (e^{t-2} \left (-2 e+14 e^2+e^t-8 e^{t+2}\right )+1\right ) & 1<t\leq 2 \\ -\frac {1}{2}+\frac {1}{2} e^{2 t-4} \left (-2+e^2-8 e^4\right )+e^{t-2} (2+e (-1+7 e)) & \text {True} \\ \\ \\ \\ \\ \end{align*}