56.1.76 problem 76

Internal problem ID [8788]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 76
Date solved : Sunday, March 30, 2025 at 01:35:49 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 107

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

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

✓ Sympy. Time used: 0.163 (sec). Leaf size: 94

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

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