56.1.71 problem 71

Internal problem ID [8783]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 71
Date solved : Wednesday, March 05, 2025 at 06:48:24 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=9 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-6 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=6 x \left (t \right )+4 y \left (t \right )+3 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 57

ode:=[diff(x(t),t) = 9*x(t)+4*y(t), diff(y(t),t) = -6*x(t)-y(t), diff(z(t),t) = 6*x(t)+4*y(t)+3*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{5 t}-\frac {3 c_3 \,{\mathrm e}^{3 t}}{2} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{3 t}+c_{1} {\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 103

ode={D[x[t],t]==9*x[t]+4*y[t],D[y[t],t]==-6*x[t]-y[t],D[z[t],t]==6*x[t]+4*y[t]+3*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{3 t} \left (c_1 \left (3 e^{2 t}-2\right )+2 c_2 \left (e^{2 t}-1\right )\right ) \\ y(t)\to -e^{3 t} \left (3 c_1 \left (e^{2 t}-1\right )+c_2 \left (2 e^{2 t}-3\right )\right ) \\ z(t)\to e^{3 t} \left (3 c_1 \left (e^{2 t}-1\right )+2 c_2 \left (e^{2 t}-1\right )+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-9*x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(6*x(t) + y(t) + Derivative(y(t), t),0),Eq(-6*x(t) - 4*y(t) - 3*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{3 t}}{3} + C_{2} e^{5 t}, \ y{\left (t \right )} = C_{1} e^{3 t} - C_{2} e^{5 t}, \ z{\left (t \right )} = C_{2} e^{5 t} + C_{3} e^{3 t}\right ] \]