15.17.7 problem 12

Internal problem ID [3243]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 26, page 115
Problem number : 12
Date solved : Sunday, March 30, 2025 at 01:23:41 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=3 y \left (t \right )-2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.169 (sec). Leaf size: 51

ode:=[diff(x(t),t) = 3*x(t), diff(y(t),t) = 2*x(t)+3*y(t), diff(z(t),t) = 3*y(t)-2*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= \left (2 c_3 t +c_2 \right ) {\mathrm e}^{3 t} \\ z \left (t \right ) &= \left (\frac {3 \,{\mathrm e}^{5 t} \left (10 c_3 t +5 c_2 -2 c_3 \right )}{25}+c_1 \right ) {\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 78

ode={D[x[t],t]==3*x[t],D[y[t],t]==2*x[t]+3*y[t],D[z[t],t]==3*y[t]-2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 e^{3 t} \\ y(t)\to e^{3 t} (2 c_1 t+c_2) \\ z(t)\to \frac {1}{25} e^{-2 t} \left (6 c_1 \left (e^{5 t} (5 t-1)+1\right )+5 \left (3 c_2 \left (e^{5 t}-1\right )+5 c_3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.157 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 3*y(t) + Derivative(y(t), t),0),Eq(-3*y(t) + 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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