67.4.20 problem Problem 3(f)

Internal problem ID [14064]
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)
Date solved : Tuesday, January 28, 2025 at 06:13:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=36 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Solution by Maple

Time used: 10.125 (sec). Leaf size: 49

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)
 

\[ y = 9 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2-2 t}-8 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}+\left (-6 t +5\right ) \operatorname {Heaviside}\left (t -1\right )+6 t +4 \,{\mathrm e}^{-2 t}-5 \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 64

DSolve[{D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==36*t*(UnitStep[t]-UnitStep[t-1]),{y[0]==-1,Derivative[1][y][0] ==-2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-3 t} \left (4-5 e^t\right ) & t<0 \\ e^{-3 t} \left (-8 e^3+4 e^t+9 e^{t+2}\right ) & t>1 \\ 6 t+4 e^{-2 t}-5 & \text {True} \\ \end {array} \\ \end {array} \]