61.4.1 problem Problem 2(a)

Internal problem ID [15326]
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 2(a)
Date solved : Thursday, October 02, 2025 at 10:11:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+3 y&=9 t \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+3*y(t) = 9*t; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )+3 t -2 \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 13

ode=D[y[t],{t,2}]+2*D[y[t],{t,2}]+3*y[t]==9*t; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 3 t-2 \sin (t) \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*t + 3*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 3 t - 2 + 2 e^{- t} \cos {\left (\sqrt {2} t \right )} \]