60.9.19 problem 1874
Internal
problem
ID
[11798]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1874
Date
solved
:
Sunday, March 30, 2025 at 09:15:38 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right ) f \left (t \right )+y \left (t \right ) g \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right ) g \left (t \right )+y \left (t \right ) f \left (t \right ) \end{align*}
✓ Maple. Time used: 0.554 (sec). Leaf size: 56
ode:=[diff(x(t),t) = x(t)*f(t)+y(t)*g(t), diff(y(t),t) = -x(t)*g(t)+y(t)*f(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\int \left (\tan \left (c_1 -\int g \left (t \right )d t \right ) g \left (t \right )+f \left (t \right )\right )d t} c_2 \\
y \left (t \right ) &= {\mathrm e}^{\int \left (\tan \left (c_1 -\int g \left (t \right )d t \right ) g \left (t \right )+f \left (t \right )\right )d t} c_2 \tan \left (c_1 -\int g \left (t \right )d t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 93
ode={D[x[t],t]==x[t]*f[t]+y[t]*g[t],D[y[t],t]==-x[t]*g[t]+y[t]*f[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \exp \left (\int _1^tf(K[2])dK[2]\right ) \left (c_1 \cos \left (\int _1^tg(K[1])dK[1]\right )+c_2 \sin \left (\int _1^tg(K[1])dK[1]\right )\right ) \\
y(t)\to \exp \left (\int _1^tf(K[2])dK[2]\right ) \left (c_2 \cos \left (\int _1^tg(K[1])dK[1]\right )-c_1 \sin \left (\int _1^tg(K[1])dK[1]\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.453 (sec). Leaf size: 92
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-f(t)*x(t) - g(t)*y(t) + Derivative(x(t), t),0),Eq(-f(t)*y(t) + g(t)*x(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {i C_{2}}{2}\right ) e^{\int f{\left (t \right )}\, dt + i \int g{\left (t \right )}\, dt} + \left (\frac {C_{1}}{2} + \frac {i C_{2}}{2}\right ) e^{\int f{\left (t \right )}\, dt - i \int g{\left (t \right )}\, dt}, \ y{\left (t \right )} = - \left (\frac {i C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{\int f{\left (t \right )}\, dt - i \int g{\left (t \right )}\, dt} + \left (\frac {i C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\int f{\left (t \right )}\, dt + i \int g{\left (t \right )}\, dt}\right ]
\]