Internal problem ID [2876]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for
10.7. page 704
Problem number: Problem 36.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-4 y=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 4] \end {align*}
✓ Solution by Maple
Time used: 2.579 (sec). Leaf size: 35
dsolve([diff(y(t),t$2)-4*y(t)=Heaviside(t-1)-Heaviside(t-2),y(0) = 0, D(y)(0) = 4],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -1\right ) \sinh \left (t -1\right )^{2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) \sinh \left (t -2\right )^{2}}{2}+2 \sinh \left (2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 113
DSolve[{y''[t]-4*y[t]==UnitStep[t-1]-UnitStep[t-2],{y[0]==0,y'[0]==4}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} \left (-1+e^{4 t}\right ) & t\leq 1 \\ \frac {1}{8} \left (-2+e^{2-2 t}-8 e^{-2 t}+8 e^{2 t}+e^{2 t-2}\right ) & 1<t\leq 2 \\ \frac {1}{8} e^{-2 (t+2)} \left (-8 e^4+e^6-e^8-e^{4 t}+e^{4 t+2}+8 e^{4 t+4}\right ) & \text {True} \\ \end {array} \\ \end {array} \]