83.15.9 problem 4 (i)

Internal problem ID [22033]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 4 (i)
Date solved : Thursday, October 02, 2025 at 08:21:50 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime }&=0 \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 )&=8 \\ y^{\prime \prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 15

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-4*diff(diff(y(t),t),t) = 0; 
ic:=[y(0) = 5, D(y)(0) = 2, (D@@2)(y)(0) = 8, (D@@3)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 t +3+2 \cosh \left (2 t \right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 20

ode=D[y[t],{t,4}]-4*D[y[t],{t,2}]==0; 
ic={y[0]==5,Derivative[1][y][0] ==2,Derivative[2][y][0] ==8,Derivative[3][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 t+e^{-2 t}+e^{2 t}+3 \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = 2 t + \frac {e^{2 t}}{4} + \frac {9}{2} + \frac {e^{- 2 t}}{4} \]