61.4.21 problem Problem 3(g)

Internal problem ID [15346]
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(g)
Date solved : Thursday, October 02, 2025 at 10:11:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=39 \operatorname {Heaviside}\left (t \right )-507 \left (t -2\right ) \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.347 (sec). Leaf size: 64

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+13*y(t) = 39*Heaviside(t)-507*(t-2)*Heaviside(t-2); 
ic:=[y(0) = 3, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 3-12 \left (\left (-\frac {5 \cos \left (6\right )}{12}+\sin \left (6\right )\right ) \sin \left (3 t \right )+\cos \left (3 t \right ) \left (\cos \left (6\right )+\frac {5 \sin \left (6\right )}{12}\right )\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{4-2 t}+3 \left (30-13 t \right ) \operatorname {Heaviside}\left (t -2\right )+\frac {{\mathrm e}^{-2 t} \sin \left (3 t \right )}{3} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 103

ode=D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==39*UnitStep[t]-507*(t-2)*UnitStep[t-2]; 
ic={y[0]==3,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} -39 t-12 e^{4-2 t} \cos (6-3 t)-5 e^{4-2 t} \sin (6-3 t)+\frac {1}{3} e^{-2 t} \sin (3 t)+93 & t>2 \\ \frac {1}{3} e^{-2 t} \sin (3 t)+3 & 0\leq t\leq 2 \\ \frac {1}{3} e^{-2 t} (9 \cos (3 t)+7 \sin (3 t)) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 2.112 (sec). Leaf size: 99

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((507*t - 1014)*Heaviside(t - 2) + 13*y(t) - 39*Heaviside(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 39 t \theta \left (t - 2\right ) + \left (\left (\frac {7}{3} - 2 \theta \left (t\right )\right ) \sin {\left (3 t \right )} + \left (3 - 3 \theta \left (t\right )\right ) \cos {\left (3 t \right )} + 5 e^{4} \sin {\left (3 t - 6 \right )} \theta \left (t - 2\right ) - 12 e^{4} \cos {\left (3 t - 6 \right )} \theta \left (t - 2\right )\right ) e^{- 2 t} + 3 \theta \left (t\right ) + 90 \theta \left (t - 2\right ) \]