67.5.18 problem Problem 3(g)

Internal problem ID [14033]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 3(g)
Date solved : Monday, March 31, 2025 at 08:22:31 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 163

ode:=[diff(x(t),t) = -1/2*x(t)+2*y(t)-3*z(t), diff(y(t),t) = y(t)-1/2*z(t), diff(z(t),t) = -2*x(t)+z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {c_3 \,{\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}-c_1 \,{\mathrm e}^{3 t}+\frac {7 c_2 \,{\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}}{8}+\frac {7 c_3 \,{\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}}}{8} \\ y \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}-\frac {c_3 \,{\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}-\frac {c_1 \,{\mathrm e}^{3 t}}{4}+\frac {7 c_2 \,{\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}}{8}+\frac {7 c_3 \,{\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}}}{8} \\ z \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}+c_3 \,{\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \\ \end{align*}

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 523

ode={D[x[t],t]==-1/2*x[t]+2*y[t]-3*z[t],D[y[t],t]==y[t]-1/2*z[t],D[z[t],t]==-2*x[t]+z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{264} e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (c_1 \left (\left (88-16 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}+88 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+88+16 \sqrt {33}\right )+4 c_2 \left (\left (3 \sqrt {33}-11\right ) e^{\frac {\sqrt {33} t}{2}}+22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-11-3 \sqrt {33}\right )-c_3 \left (\left (13 \sqrt {33}-77\right ) e^{\frac {\sqrt {33} t}{2}}+154 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-77-13 \sqrt {33}\right )\right ) \\ y(t)\to \frac {e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (-4 c_1 \left (\left (11+5 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}-22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+11-5 \sqrt {33}\right )+c_2 \left (\left (484+92 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}+88 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+484-92 \sqrt {33}\right )+c_3 \left (\left (77+3 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}-154 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+77-3 \sqrt {33}\right )\right )}{1056} \\ z(t)\to -\frac {1}{264} e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (4 c_1 \left (\left (3 \sqrt {33}-11\right ) e^{\frac {\sqrt {33} t}{2}}+22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-11-3 \sqrt {33}\right )-4 c_2 \left (\left (11+5 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}-22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+11-5 \sqrt {33}\right )+c_3 \left (\left (7 \sqrt {33}-55\right ) e^{\frac {\sqrt {33} t}{2}}-154 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-55-7 \sqrt {33}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.326 (sec). Leaf size: 143

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

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (7 - \sqrt {33}\right ) e^{- \frac {t \left (3 - \sqrt {33}\right )}{4}}}{8} + \frac {C_{2} \left (\sqrt {33} + 7\right ) e^{- \frac {t \left (3 + \sqrt {33}\right )}{4}}}{8} - C_{3} e^{3 t}, \ y{\left (t \right )} = \frac {C_{1} \left (\sqrt {33} + 7\right ) e^{- \frac {t \left (3 - \sqrt {33}\right )}{4}}}{8} + \frac {C_{2} \left (7 - \sqrt {33}\right ) e^{- \frac {t \left (3 + \sqrt {33}\right )}{4}}}{8} - \frac {C_{3} e^{3 t}}{4}, \ z{\left (t \right )} = C_{1} e^{- \frac {t \left (3 - \sqrt {33}\right )}{4}} + C_{2} e^{- \frac {t \left (3 + \sqrt {33}\right )}{4}} + C_{3} e^{3 t}\right ] \]