87.22.32 problem 32

Internal problem ID [23777]
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 : 32
Date solved : Thursday, October 02, 2025 at 09:45:05 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+y&=18 \,{\mathrm e}^{2 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=13 \\ y^{\prime \prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 30

ode:=diff(diff(diff(y(t),t),t),t)+y(t) = 18*exp(2*t); 
ic:=[y(0) = -1, D(y)(0) = 13, (D@@2)(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -7 \,{\mathrm e}^{-t}+4 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+2 \,{\mathrm e}^{2 t} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 40

ode=D[y[t],{t,3}]+y[t]==18*Exp[2*t]; 
ic={y[0]==-1,Derivative[1][y][0] ==13,Derivative[2][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \left (2 e^{3 t}+4 e^{3 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )-7\right ) \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 18*exp(2*t) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 13, Subs(Derivative(y(t), (t, 2)), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 4 e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} + 2 e^{2 t} - 7 e^{- t} \]