83.15.11 problem 4 (k)

Internal problem ID [22035]
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 (k)
Date solved : Thursday, October 02, 2025 at 08:21:51 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=3 \\ y^{\prime \prime }\left (0\right )&=-4 \\ y^{\prime \prime \prime }\left (0\right )&=12 \\ \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 23

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = 3, (D@@2)(y)(0) = -4, (D@@3)(y)(0) = 12]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {9 \sin \left (t \right )}{2}+\frac {11 \,{\mathrm e}^{t}}{4}-\frac {19 \,{\mathrm e}^{-t}}{4}+2 \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 30

ode=D[y[t],{t,4}]-y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==3,Derivative[2][y][0] ==-4,Derivative[3][y][0] ==12}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{4} \left (-19 e^{-t}+11 e^t-18 \sin (t)+8 \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.065 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 3, Subs(Derivative(y(t), (t, 2)), t, 0): -4, Subs(Derivative(y(t), (t, 3)), t, 0): 12} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {11 e^{t}}{4} - \frac {9 \sin {\left (t \right )}}{2} + 2 \cos {\left (t \right )} - \frac {19 e^{- t}}{4} \]