72.14.2 problem 4

Internal problem ID [14786]
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 : 4
Date solved : Thursday, March 13, 2025 at 04:18:58 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 76

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

\begin{align*} x(t)\to c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to c_2 \cos (t)-c_1 \sin (t) \\ z(t)\to c_3 e^{2 t} \\ x(t)\to c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to c_2 \cos (t)-c_1 \sin (t) \\ z(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.081 (sec). Leaf size: 32

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

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