60.9.51 problem 1907

Internal problem ID [11830]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1907
Date solved : Wednesday, March 05, 2025 at 03:07:36 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )+48 y \left (t \right )-28 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )+40 y \left (t \right )-22 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-6 x \left (t \right )+57 y \left (t \right )-31 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 65

ode:=[diff(x(t),t) = -3*x(t)+48*y(t)-28*z(t), diff(y(t),t) = -4*x(t)+40*y(t)-22*z(t), diff(z(t),t) = -6*x(t)+57*y(t)-31*z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 157

ode={D[x[t],t]==-3*x[t]+48*y[t]-28*z[t],D[y[t],t]==-4*x[t]+40*y[t]-22*z[t],D[z[t],t]==-6*x[t]+57*y[t]-31*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^t \left (c_1 \left (3-2 e^{2 t}\right )+2 \left (e^t-1\right ) \left (3 c_2 \left (3 e^t+5\right )-c_3 \left (5 e^t+9\right )\right )\right ) \\ y(t)\to e^t \left (-2 c_1 \left (e^{2 t}-1\right )+c_2 \left (3 e^t+18 e^{2 t}-20\right )-2 c_3 \left (e^t+5 e^{2 t}-6\right )\right ) \\ z(t)\to e^t \left (-3 c_1 \left (e^{2 t}-1\right )+3 c_2 \left (e^t+9 e^{2 t}-10\right )-c_3 \left (2 e^t+15 e^{2 t}-18\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(3*x(t) - 48*y(t) + 28*z(t) + Derivative(x(t), t),0),Eq(4*x(t) - 40*y(t) + 22*z(t) + Derivative(y(t), t),0),Eq(6*x(t) - 57*y(t) + 31*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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