Internal problem ID [10980]
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 3(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y-36 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (-1+t \right )\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = -2] \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 67
dsolve([diff(y(t),t$2)+5*diff(y(t),t)+6*y(t)=36*t*(Heaviside(t)-Heaviside(t-1)),y(0) = -1, D(y)(0) = -2],y(t), singsol=all)
\[ y \left (t \right ) = 6 \left (\left (\left (-t +\frac {5}{6}\right ) {\mathrm e}^{3 t}-\frac {4 \,{\mathrm e}^{3}}{3}+\frac {3 \,{\mathrm e}^{t +2}}{2}\right ) \operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t \right ) \left (t -\frac {5}{6}\right ) {\mathrm e}^{3 t}+\left (\frac {3 \,{\mathrm e}^{t}}{2}-\frac {2}{3}\right ) \operatorname {Heaviside}\left (t \right )-\frac {5 \,{\mathrm e}^{t}}{6}+\frac {2}{3}\right ) {\mathrm e}^{-3 t} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 63
DSolve[{y''[t]+5*y'[t]+6*y[t]==36*t*(UnitStep[t]-UnitStep[t-1]),{y[0]==-1,y'[0]==-2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} e^{-3 t} \left (4-5 e^t\right ) & t<0 \\ e^{-3 t} \left (-8 e^3+e^t \left (4+9 e^2\right )\right ) & t>1 \\ 6 t+4 e^{-2 t}-5 & \text {True} \\ \\ \\ \\ \\ \end{align*}