67.3.26 problem Problem 27

Internal problem ID [13973]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 27
Date solved : Monday, March 31, 2025 at 08:20:36 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+13 y^{\prime \prime }+36 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=-1\\ y^{\prime \prime }\left (0\right )&=5\\ y^{\prime \prime \prime }\left (0\right )&=19 \end{align*}

✓ Maple. Time used: 0.150 (sec). Leaf size: 25

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+13*diff(diff(y(t),t),t)+36*y(t) = 0; 
ic:=y(0) = 0, D(y)(0) = -1, (D@@2)(y)(0) = 5, (D@@3)(y)(0) = 19; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \cos \left (2 t \right )+\sin \left (2 t \right )-\cos \left (3 t \right )-\sin \left (3 t \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 26

ode=D[y[t],{t,4}]+13*D[y[t],{t,2}]+36*y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==-1,Derivative[2][y][0] ==5,Derivative[3][y][0]==19}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \sin (2 t)-\sin (3 t)+\cos (2 t)-\cos (3 t) \]

✓ Sympy. Time used: 0.155 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(36*y(t) + 13*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -1, Subs(Derivative(y(t), (t, 2)), t, 0): 5, Subs(Derivative(y(t), (t, 3)), t, 0): 19} 
dsolve(ode,func=y(t),ics=ics)
 

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