60.9.44 problem 1899

Internal problem ID [11823]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1899
Date solved : Sunday, March 30, 2025 at 09:16:02 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.158 (sec). Leaf size: 51

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

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {3 c_3 \,{\mathrm e}^{2 t}}{4}+c_2 \,{\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}-\frac {3 c_3 \,{\mathrm e}^{2 t}}{2}-\frac {2 c_2 \,{\mathrm e}^{-2 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 93

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

\begin{align*} x(t)\to c_1 e^{2 t} \\ y(t)\to \frac {1}{4} e^{-2 t} \left (3 c_1 \left (e^{4 t}-1\right )+4 c_2\right ) \\ z(t)\to \frac {1}{10} e^{-2 t} \left (c_1 \left (-15 e^{4 t}+12 e^{5 t}+3\right )+4 c_2 \left (e^{5 t}-1\right )+10 c_3 e^{5 t}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 58

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

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