67.4.20 problem Problem 3(f)

Internal problem ID [13993]
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 : Monday, March 31, 2025 at 08:21:05 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*}

✓ Maple. Time used: 0.163 (sec). Leaf size: 49

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+6*y(t) = 36*t*(Heaviside(t)-Heaviside(t-1)); 
ic:=y(0) = -1, D(y)(0) = -2; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 1.531 (sec). Leaf size: 87

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-36*t*(Heaviside(t) - Heaviside(t - 1)) + 6*y(t) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

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