Internal problem ID [2367]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-4 y-\theta \left (t -1\right )+\theta \left (t -2\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 4] \end {align*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 75
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)
\[ y \relax (t ) = -{\mathrm e}^{-2 t}+{\mathrm e}^{2 t}-\frac {\theta \left (t -1\right )}{4}+\frac {\theta \left (t -1\right ) {\mathrm e}^{-2 t +2}}{8}+\frac {\theta \left (t -2\right )}{4}-\frac {\theta \left (t -2\right ) {\mathrm e}^{-2 t +4}}{8}+\frac {\theta \left (t -1\right ) {\mathrm e}^{2 t -2}}{8}-\frac {\theta \left (t -2\right ) {\mathrm e}^{2 t -4}}{8} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 61
DSolve[{y''[t]-4*y[t]==UnitStep[t-1]-UnitStep[t-2],{y[0]==0,y'[0]==4}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{4} (-\theta (1-t) (\cosh (2-2 t)-1)+\theta (2-t) (\cosh (4-2 t)-1)+8 \sinh (2 t)+\cosh (2-2 t)-\cosh (4-2 t)) \\ \end{align*}