67.3.21 problem Problem 22

Internal problem ID [13968]
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 22
Date solved : Monday, March 31, 2025 at 08:20:29 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+6 y^{\prime \prime }+13 y^{\prime }&=0 \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 )&=-6 \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 16

ode:=diff(diff(diff(y(t),t),t),t)+6*diff(diff(y(t),t),t)+13*diff(y(t),t) = 0; 
ic:=y(0) = 1, D(y)(0) = 1, (D@@2)(y)(0) = -6; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-3 t} \sin \left (2 t \right )}{2}+1 \]

✓ Mathematica. Time used: 60.116 (sec). Leaf size: 73

ode=D[ y[t],{t,3}]+6*D[y[t],{t,2}]+13*D[y[t],t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==1,Derivative[2][y][0] ==-6}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t\frac {1}{2} e^{-3 K[1]} (2 \cos (2 K[1])-3 \sin (2 K[1]))dK[1]-\int _1^0\frac {1}{2} e^{-3 K[1]} (2 \cos (2 K[1])-3 \sin (2 K[1]))dK[1]+1 \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(y(t), (t, 2)), t, 0): -6} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 1 + \frac {e^{- 3 t} \sin {\left (2 t \right )}}{2} \]