77.1.164 problem 191 (page 298)

Internal problem ID [17975]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 191 (page 298)
Date solved : Friday, March 14, 2025 at 04:53:34 AM
CAS classification : system_of_ODEs

\begin{align*} t \left (\frac {d}{d t}x \left (t \right )\right )+6 x \left (t \right )-y \left (t \right )-3 z \left (t \right )&=0\\ t \left (\frac {d}{d t}y \left (t \right )\right )+23 x \left (t \right )-6 y \left (t \right )-9 z \left (t \right )&=0\\ t \left (\frac {d}{d t}z \left (t \right )\right )+x \left (t \right )+y \left (t \right )-2 z \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 63

ode:=[t*diff(x(t),t)+6*x(t)-y(t)-3*z(t) = 0, t*diff(y(t),t)+23*x(t)-6*y(t)-9*z(t) = 0, t*diff(z(t),t)+x(t)+y(t)-2*z(t) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 66

ode={t*D[x[t],t]+6*x[t]-y[t]-3*z[t]==0,t*D[y[t],t]+23*x[t]-6*y[t]-9*z[t]==0,t*D[z[t],t]+x[t]+y[t]-2*z[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.155 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(t*Derivative(x(t), t) + 6*x(t) - y(t) - 3*z(t),0),Eq(t*Derivative(y(t), t) + 23*x(t) - 6*y(t) - 9*z(t),0),Eq(t*Derivative(z(t), t) + x(t) + y(t) - 2*z(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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