60.9.37 problem 1892
Internal
problem
ID
[11816]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1892
Date
solved
:
Friday, March 14, 2025 at 02:58:27 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )-a \left (\frac {d}{d t}y \left (t \right )\right )+b x \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+a \left (\frac {d}{d t}x \left (t \right )\right )+b y \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.132 (sec). Leaf size: 867
ode:=[diff(diff(x(t),t),t)-a*diff(y(t),t)+b*x(t) = 0, diff(diff(y(t),t),t)+a*diff(x(t),t)+b*y(t) = 0];
dsolve(ode);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.16 (sec). Leaf size: 3522
ode={D[x[t],{t,2}]-a*D[y[t],t]+b*x[t]==0,D[y[t],{t,2}]+a*D[x[t],t]+b*y[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 1.330 (sec). Leaf size: 450
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
x = Function("x")
y = Function("y")
ode=[Eq(-a*Derivative(y(t), t) + b*x(t) + Derivative(x(t), (t, 2)),0),Eq(a*Derivative(x(t), t) + b*y(t) + Derivative(y(t), (t, 2)),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {2 C_{1} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a - \sqrt {a^{2} + 4 b}} - \frac {2 C_{2} e^{\frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a - \sqrt {a^{2} + 4 b}} - \frac {2 C_{3} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a + \sqrt {a^{2} + 4 b}} - \frac {2 C_{4} e^{\frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{a + \sqrt {a^{2} + 4 b}}, \ y{\left (t \right )} = - \frac {\sqrt {2} C_{1} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}} + \frac {\sqrt {2} C_{2} e^{\frac {\sqrt {2} t \sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} + a \sqrt {a^{2} + 4 b} - 2 b}} - \frac {\sqrt {2} C_{3} e^{- \frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}} + \frac {\sqrt {2} C_{4} e^{\frac {\sqrt {2} t \sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}{2}}}{\sqrt {- a^{2} - a \sqrt {a^{2} + 4 b} - 2 b}}\right ]
\]