87.22.19 problem 19

Internal problem ID [23764]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:44:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=27 t \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+9*y(t) = 27*t; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (6 t +3\right ) {\mathrm e}^{-3 t}+3 t -2 \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 21

ode=D[y[t],{t,2}]+6*D[y[t],{t,1}]+9*y[t]==27*t; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 3 t+e^{-3 t} (6 t+3)-2 \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-27*t + 9*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 3 t + \left (6 t + 3\right ) e^{- 3 t} - 2 \]