28.3.19 problem 6.54

Internal problem ID [4532]
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 : 6.54
Date solved : Tuesday, March 04, 2025 at 06:51:50 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y&=\left (2 t^{2}+t +1\right ) \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-2\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 3.913 (sec). Leaf size: 46

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+4*y(t) = (2*t^2+t+1)*Dirac(t-1); 
ic:=y(0) = 1, D(y)(0) = -2, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\sinh \left (t \right ) \cos \left (t \right )+\left (\cos \left (t \right )-\sin \left (t \right )\right ) \cosh \left (t \right )+\left (-\cos \left (-1+t \right ) \sinh \left (-1+t \right )+\sin \left (-1+t \right ) \cosh \left (-1+t \right )\right ) \operatorname {Heaviside}\left (-1+t \right ) \]

✓ Mathematica. Time used: 0.203 (sec). Leaf size: 75

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

\[ y(t)\to \frac {1}{4} e^{-t-1} \left (-2 \theta (t-1) \left (\left (e^{2 t}+e^2\right ) \sin (1-t)+\left (e^{2 t}-e^2\right ) \cos (1-t)\right )-2 e \left (e^{2 t}+1\right ) \sin (t)+4 e \cos (t)\right ) \]

✓ Sympy. Time used: 43.405 (sec). Leaf size: 466

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-2*t**2 - t - 1)*Dirac(t - 1) + 4*y(t) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -2, 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)
 

\[ \text {Solution too large to show} \]