7.22.10 problem 20

Internal problem ID [585]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.1 (First order systems and applications). Problems at page 335
Problem number : 20
Date solved : Saturday, March 29, 2025 at 04:57:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-9 x \left (t \right )+6 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 31

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -9*x(t)+6*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= {\mathrm e}^{3 t} \left (3 c_2 t +3 c_1 +c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 45

ode={D[x[t],t]==y[t],D[y[t],t]==-9*x[t]+6*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{3 t} (-3 c_1 t+c_2 t+c_1) \\ y(t)\to e^{3 t} (-9 c_1 t+3 c_2 t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(9*x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 C_{2} t e^{3 t} - \left (3 C_{1} - C_{2}\right ) e^{3 t}, \ y{\left (t \right )} = - 9 C_{1} e^{3 t} - 9 C_{2} t e^{3 t}\right ] \]