60.9.45 problem 1900

Internal problem ID [11824]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1900
Date solved : Wednesday, March 05, 2025 at 03:07:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )-4 y \left (t \right )+z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 49

ode:=[diff(x(t),t) = 4*x(t), diff(y(t),t) = x(t)-2*y(t), diff(z(t),t) = x(t)-4*y(t)+z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{4 t} \\ y \left (t \right ) &= \frac {c_3 \,{\mathrm e}^{4 t}}{6}+{\mathrm e}^{-2 t} c_{2} \\ z &= \frac {c_3 \,{\mathrm e}^{4 t}}{9}+c_{1} {\mathrm e}^{t}+\frac {4 \,{\mathrm e}^{-2 t} c_{2}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 88

ode={D[x[t],t]==4*x[t],D[y[t],t]==x[t]-2*y[t],D[z[t],t]==x[t]-4*y[t]+z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 e^{4 t} \\ y(t)\to \frac {1}{6} e^{-2 t} \left (c_1 \left (e^{6 t}-1\right )+6 c_2\right ) \\ z(t)\to \frac {1}{9} e^{-2 t} \left (c_1 \left (e^{3 t}+e^{6 t}-2\right )-12 c_2 \left (e^{3 t}-1\right )+9 c_3 e^{3 t}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-4*x(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + Derivative(y(t), t),0),Eq(-x(t) + 4*y(t) - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 9 C_{1} e^{4 t}, \ y{\left (t \right )} = \frac {3 C_{1} e^{4 t}}{2} + \frac {3 C_{2} e^{- 2 t}}{4}, \ z{\left (t \right )} = C_{1} e^{4 t} + C_{2} e^{- 2 t} + C_{3} e^{t}\right ] \]