61.3.22 problem Problem 23

Internal problem ID [15321]
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 23
Date solved : Thursday, October 02, 2025 at 10:11:37 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: 0.103 (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,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 60.064 (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]
 

\begin{align*} 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 \end{align*}

✓ Sympy. Time used: 0.135 (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)
 

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