85.70.7 problem 4

Internal problem ID [22931]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 6. Solution of linear differential equations by Laplace transform. A Exercises at page 283
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:16:43 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.109 (sec). Leaf size: 24

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-y(t) = cos(t); 
ic:=[y(0) = 1, D(y)(0) = -1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {3 \cosh \left (t \right )}{4}-\frac {\sinh \left (t \right )}{2}+\frac {\cos \left (t \right )}{4}-\frac {\sin \left (t \right ) \left (2+t \right )}{4} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 31

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

\begin{align*} y(t)&\to \frac {1}{8} \left (5 e^{-t}+e^t-2 (t+2) \sin (t)+2 \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = \left (- \frac {t}{4} - \frac {1}{2}\right ) \sin {\left (t \right )} + \frac {e^{t}}{8} + \frac {\cos {\left (t \right )}}{4} + \frac {5 e^{- t}}{8} \]