22.3.16 problem 6.51

Internal problem ID [4529]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.51
Date solved : Tuesday, September 30, 2025 at 07:33:56 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y&=10 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=-2 \\ y^{\prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.130 (sec). Leaf size: 23

ode:=diff(diff(diff(y(t),t),t),t)-diff(diff(y(t),t),t)+4*diff(y(t),t)-4*y(t) = 10*exp(-t); 
ic:=[y(0) = 5, D(y)(0) = -2, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \cos \left (2 t \right )-4 \sin \left (2 t \right )+4 \cosh \left (t \right )+6 \sinh \left (t \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 28

ode=D[y[t],{t,3}]-D[y[t],{t,2}]+4*D[y[t],t]-4*y[t]==10*Exp[-t]; 
ic={y[0]==5,Derivative[1][y][0] == -2,Derivative[2][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{-t}+5 e^t-4 \sin (2 t)+\cos (2 t) \end{align*}

✓ Sympy. Time used: 0.149 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) + 4*Derivative(y(t), t) - Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)) - 10*exp(-t),0) 
ics = {y(0): 5, Subs(Derivative(y(t), t), t, 0): -2, Subs(Derivative(y(t), (t, 2)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 5 e^{t} - 4 \sin {\left (2 t \right )} + \cos {\left (2 t \right )} - e^{- t} \]