72.14.9 problem 13

Internal problem ID [14793]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.8 page 371
Problem number : 13
Date solved : Thursday, March 13, 2025 at 04:19:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 30

ode:=[diff(x(t),t) = -x(t)+2*y(t), diff(y(t),t) = 2*x(t)-4*y(t), diff(z(t),t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-5 t} \\ y &= -2 c_{2} {\mathrm e}^{-5 t}+\frac {c_{1}}{2} \\ z &= c_{3} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 77

ode={D[x[t],t]==-1*x[t]+2*y[t]+0*z[t],D[y[t],t]==2*x[t]-4*y[t]+0*z[t],D[z[t],t]==0*x[t]+0*y[t]+0*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-5 t} \left (c_1 \left (4 e^{5 t}+1\right )+2 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-5 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \\ z(t)\to c_3 \\ \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 27

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

\[ \left [ x{\left (t \right )} = 2 C_{1} - \frac {C_{2} e^{- 5 t}}{2}, \ y{\left (t \right )} = C_{1} + C_{2} e^{- 5 t}, \ z{\left (t \right )} = C_{3}\right ] \]