78.4.8 problem 4.c

Internal problem ID [21036]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 5, Laplace transforms. Problems section 5.7
Problem number : 4.c
Date solved : Thursday, October 02, 2025 at 07:01:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+18 y&=2 \operatorname {Heaviside}\left (\pi -t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.353 (sec). Leaf size: 54

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+18*y(t) = 2*Heaviside(Pi-t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {1}{9}-\frac {\operatorname {Heaviside}\left (-\pi +t \right )}{9}-\frac {\left (\sin \left (3 t \right )+\cos \left (3 t \right )\right ) {\mathrm e}^{-3 t}}{9}-\frac {{\mathrm e}^{-3 t +3 \pi } \operatorname {Heaviside}\left (-\pi +t \right ) \left (\sin \left (3 t \right )+\cos \left (3 t \right )\right )}{9} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 63

ode=D[y[t],{t,2}]+6*D[y[t],t]+18*y[t]==2*UnitStep[Pi-t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{9} e^{-3 t} \left (-\cos (3 t)+e^{3 t}-\sin (3 t)\right ) & t\leq \pi \\ -\frac {1}{9} e^{-3 t} \left (1+e^{3 \pi }\right ) (\cos (3 t)+\sin (3 t)) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 1.239 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(18*y(t) - 2*Heaviside(pi - t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {e^{3 \pi }}{9} - \frac {1}{9}\right ) \sin {\left (3 t \right )} + \frac {\sqrt {2} e^{3 \pi } \sin {\left (3 t + \frac {\pi }{4} \right )} \theta \left (\pi - t\right )}{9} + \left (- \frac {e^{3 \pi }}{9} - \frac {1}{9}\right ) \cos {\left (3 t \right )}\right ) e^{- 3 t} + \frac {\theta \left (\pi - t\right )}{9} \]