87.22.33 problem 33

Internal problem ID [23778]
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 : 33
Date solved : Thursday, October 02, 2025 at 09:45:05 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+8 y&=-12 \,{\mathrm e}^{-2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-8 \\ y^{\prime }\left (0\right )&=24 \\ y^{\prime \prime }\left (0\right )&=-46 \\ \end{align*}

✓ Maple. Time used: 0.110 (sec). Leaf size: 24

ode:=diff(diff(diff(y(t),t),t),t)+8*y(t) = -12*exp(-2*t); 
ic:=[y(0) = -8, D(y)(0) = 24, (D@@2)(y)(0) = -46]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 3 \,{\mathrm e}^{t} \cos \left (\sqrt {3}\, t \right )-{\mathrm e}^{-2 t} \left (11+t \right ) \]

✓ Mathematica. Time used: 0.255 (sec). Leaf size: 30

ode=D[y[t],{t,3}]+8*y[t]==-12*Exp[-2*t]; 
ic={y[0]==-8,Derivative[1][y][0] ==24,Derivative[2][y][0] ==-46}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{-2 t} \left (t-3 e^{3 t} \cos \left (\sqrt {3} t\right )+11\right ) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(8*y(t) + Derivative(y(t), (t, 3)) + 12*exp(-2*t),0) 
ics = {y(0): -8, Subs(Derivative(y(t), t), t, 0): 24, Subs(Derivative(y(t), (t, 2)), t, 0): -46} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- t - 11\right ) e^{- 2 t} + 3 e^{t} \cos {\left (\sqrt {3} t \right )} \]