72.10.6 problem 6

Internal problem ID [14738]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.2. page 277
Problem number : 6
Date solved : Thursday, March 13, 2025 at 04:17:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=9 x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 5*x(t)+4*y(t), diff(y(t),t) = 9*x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{9 t}-\frac {4 c_{2} {\mathrm e}^{-4 t}}{9} \\ y &= c_{1} {\mathrm e}^{9 t}+c_{2} {\mathrm e}^{-4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

ode={D[x[t],t]==5*x[t]+4*y[t],D[y[t],t]==9*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{13} e^{-4 t} \left (c_1 \left (9 e^{13 t}+4\right )+4 c_2 \left (e^{13 t}-1\right )\right ) \\ y(t)\to \frac {1}{13} e^{-4 t} \left (9 c_1 \left (e^{13 t}-1\right )+c_2 \left (4 e^{13 t}+9\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(-9*x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {4 C_{1} e^{- 4 t}}{9} + C_{2} e^{9 t}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{9 t}\right ] \]