Internal problem ID [12328]
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]]
\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=36 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = -2] \end {align*}
✓ Solution by Maple
Time used: 5.203 (sec). Leaf size: 45
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)
\[ y \left (t \right ) = -8 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}+9 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}+\left (-6 t +5\right ) \operatorname {Heaviside}\left (t -1\right )+6 t +4 \,{\mathrm e}^{-2 t}-5 \]
✓ Solution by Mathematica
Time used: 0.071 (sec). Leaf size: 64
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]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-3 t} \left (4-5 e^t\right ) & t<0 \\ e^{-3 t} \left (-8 e^3+4 e^t+9 e^{t+2}\right ) & t>1 \\ 6 t+4 e^{-2 t}-5 & \text {True} \\ \end {array} \\ \end {array} \]