72.14.11 problem 15

Internal problem ID [14795]
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 : 15
Date solved : Thursday, March 13, 2025 at 04:19:06 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 )&=z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 27

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = z(t), diff(z(t),t) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 36

ode={D[x[t],t]==0*x[t]+1*y[t]+0*z[t],D[y[t],t]==0*x[t]+0*y[t]+1*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 {c_3 t^2}{2}+c_2 t+c_1 \\ y(t)\to c_3 t+c_2 \\ z(t)\to c_3 \\ \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 24

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(-z(t) + Derivative(y(t), t),0),Eq(Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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