28.3.18 problem 6.53

Internal problem ID [4531]
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 : 6.53
Date solved : Sunday, March 30, 2025 at 03:25:28 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-4 y&=40 t^{2} \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.160 (sec). Leaf size: 46

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+3*diff(diff(y(t),t),t)-4*y(t) = 40*t^2*Heaviside(t-2); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -2\right ) \left (-10 t^{2}-15+8 \,{\mathrm e}^{2-t}+40 \,{\mathrm e}^{t -2}+7 \cos \left (-4+2 t \right )+4 \sin \left (-4+2 t \right )\right ) \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 51

ode=D[y[t],{t,4}]+3*D[y[t],{t,2}]-4*y[t]==40*t^2*UnitStep[t-2]; 
ic={y[0]==0,Derivative[1][y][0] == 0,Derivative[2][y][0] == 0,Derivative[3][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -10 t^2+8 e^{2-t}+40 e^{t-2}+7 \cos (4-2 t)-4 \sin (4-2 t)-15 & t>2 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 2.093 (sec). Leaf size: 110

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-40*t**2*Heaviside(t - 2) - 4*y(t) + 3*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, 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)
 

\[ y{\left (t \right )} = - 10 t^{2} \theta \left (t - 2\right ) + \frac {40 e^{t} \theta \left (t - 2\right )}{e^{2}} - 14 \sin {\left (t \right )} \sin {\left (t - 4 \right )} \theta \left (t - 2\right ) + 8 \sin {\left (t \right )} \cos {\left (t - 4 \right )} \theta \left (t - 2\right ) - 15 \theta \left (t - 2\right ) + 7 \cos {\left (4 \right )} \theta \left (t - 2\right ) - 4 \sin {\left (4 \right )} \theta \left (t - 2\right ) + 8 e^{2} e^{- t} \theta \left (t - 2\right ) \]