60.10.9 problem 1923
Internal
problem
ID
[11844]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1923
Date
solved
:
Sunday, March 30, 2025 at 09:18:23 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \left (t^{2}+1\right ) \left (\frac {d}{d t}x \left (t \right )\right )&=-t x \left (t \right )+y \left (t \right )\\ \left (t^{2}+1\right ) \left (\frac {d}{d t}y \left (t \right )\right )&=-x \left (t \right )-t y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.107 (sec). Leaf size: 34
ode:=[(t^2+1)*diff(x(t),t) = -t*x(t)+y(t), (t^2+1)*diff(y(t),t) = -x(t)-t*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {c_1 t +c_2}{t^{2}+1} \\
y \left (t \right ) &= \frac {-c_2 t +c_1}{t^{2}+1} \\
\end{align*}
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 107
ode={(t^2+1)*D[x[t],t]==-t*x[t]+y[t],(t^2+1)*D[y[t],t]==-x[t]-t*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {c_1 \cos \left (\int _1^t\frac {1}{K[1]^2+1}dK[1]\right )+c_2 \sin \left (\int _1^t\frac {1}{K[1]^2+1}dK[1]\right )}{\sqrt {t^2+1}} \\
y(t)\to \frac {c_2 \cos \left (\int _1^t\frac {1}{K[1]^2+1}dK[1]\right )-c_1 \sin \left (\int _1^t\frac {1}{K[1]^2+1}dK[1]\right )}{\sqrt {t^2+1}} \\
\end{align*}
✓ Sympy. Time used: 0.329 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(t*x(t) + (t**2 + 1)*Derivative(x(t), t) - y(t),0),Eq(t*y(t) + (t**2 + 1)*Derivative(y(t), t) + x(t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {\left (\frac {C_{1}}{2} - \frac {i C_{2}}{2}\right ) e^{i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}} + \frac {\left (\frac {C_{1}}{2} + \frac {i C_{2}}{2}\right ) e^{- i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}}, \ y{\left (t \right )} = - \frac {\left (\frac {i C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}} + \frac {\left (\frac {i C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}}\right ]
\]