60.9.47 problem 1902
Internal
problem
ID
[11826]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1902
Date
solved
:
Sunday, March 30, 2025 at 09:16:07 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-y \left (t \right )+z \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )-y \left (t \right )&=t\\ \frac {d}{d t}z \left (t \right )-x \left (t \right )-z \left (t \right )&=t \end{align*}
✓ Maple. Time used: 0.169 (sec). Leaf size: 55
ode:=[diff(x(t),t)-y(t)+z(t) = 0, diff(y(t),t)-x(t)-y(t) = t, diff(z(t),t)-x(t)-z(t) = t];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{t} \\
y \left (t \right ) &= t c_3 \,{\mathrm e}^{t}+{\mathrm e}^{t} c_1 -c_2 -t -1 \\
z \left (t \right ) &= t c_3 \,{\mathrm e}^{t}+{\mathrm e}^{t} c_1 -c_3 \,{\mathrm e}^{t}-c_2 -t -1 \\
\end{align*}
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 246
ode={D[x[t],t]-y[t]+z[t]==0,D[y[t],t]-x[t]-y[t]==t,D[z[t],t]-x[t]-z[t]==t};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \left (e^t-1\right ) \int _1^te^{-K[1]} K[1]dK[1]-\left (e^t-1\right ) \int _1^te^{-K[2]} K[2]dK[2]+c_2 \left (e^t-1\right )-c_3 \left (e^t-1\right )+c_1 \\
y(t)\to \left (e^t t+1\right ) \int _1^te^{-K[1]} K[1]dK[1]+\left (-e^t (t-1)-1\right ) \int _1^te^{-K[2]} K[2]dK[2]+c_1 \left (e^t-1\right )+c_2 \left (e^t t+1\right )+c_3 \left (-e^t (t-1)-1\right ) \\
z(t)\to \left (e^t (t-1)+1\right ) \int _1^te^{-K[1]} K[1]dK[1]-\left (e^t (t-2)+1\right ) \int _1^te^{-K[2]} K[2]dK[2]+c_1 \left (e^t-1\right )+c_2 \left (e^t (t-1)+1\right )-c_3 \left (e^t (t-2)+1\right ) \\
\end{align*}
✓ Sympy. Time used: 0.145 (sec). Leaf size: 48
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-y(t) + z(t) + Derivative(x(t), t),0),Eq(-t - x(t) - y(t) + Derivative(y(t), t),0),Eq(-t - x(t) - z(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} + C_{2} e^{t}, \ y{\left (t \right )} = C_{1} + C_{2} t e^{t} - t + \left (C_{2} + C_{3}\right ) e^{t} - 1, \ z{\left (t \right )} = C_{1} + C_{2} t e^{t} + C_{3} e^{t} - t - 1\right ]
\]