22.3.17 problem 6.52

Internal problem ID [4530]
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 : 6.52
Date solved : Tuesday, September 30, 2025 at 07:33:56 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y&=120 \,{\mathrm e}^{3 t} \operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=15 \\ y^{\prime }\left (0\right )&=-6 \\ y^{\prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.302 (sec). Leaf size: 81

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-5*diff(diff(y(t),t),t)+4*y(t) = 120*exp(3*t)*Heaviside(t-1); 
ic:=[y(0) = 15, D(y)(0) = -6, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -10 \,{\mathrm e}^{1+2 t} \operatorname {Heaviside}\left (t -1\right )-5 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{4-t}+2 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{5-2 t}+\left (3 \,{\mathrm e}^{3 t}+10 \,{\mathrm e}^{2+t}\right ) \operatorname {Heaviside}\left (t -1\right )+6 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{-2 t}+14 \,{\mathrm e}^{-t}-2 \,{\mathrm e}^{2 t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 95

ode=D[y[t],{t,4}]-5*D[y[t],{t,2}]+4*y[t]==120*Exp[3*t]*UnitStep[t-1]; 
ic={y[0]==15,Derivative[1][y][0] == -6,Derivative[2][y][0] == 0,Derivative[3][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (-\left (\left (3 e^t+2 e\right ) \left (e-e^t\right )^4 \theta (1-t)\right )+14 e^t+6 e^{3 t}-2 e^{4 t}+3 e^{5 t}-5 e^{t+4}+10 e^{3 t+2}-10 e^{4 t+1}+2 e^5-3\right ) \end{align*}

✓ Sympy. Time used: 0.541 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 120*exp(3*t)*Heaviside(t - 1) - 5*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 15, Subs(Derivative(y(t), t), t, 0): -6, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- 10 e \theta \left (t - 1\right ) - 2\right ) e^{2 t} + \left (10 e^{2} \theta \left (t - 1\right ) + 6\right ) e^{t} + \left (- 5 e^{4} \theta \left (t - 1\right ) + 14\right ) e^{- t} + \left (2 e^{5} \theta \left (t - 1\right ) - 3\right ) e^{- 2 t} + 3 e^{3 t} \theta \left (t - 1\right ) \]