54.9.43 problem 1898
Internal
problem
ID
[13121]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1898
Date
solved
:
Sunday, October 12, 2025 at 02:28:26 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )-\frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d^{2}}{d t^{2}}y \left (t \right )-x \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.148 (sec). Leaf size: 72
ode:=[diff(diff(x(t),t),t)-diff(x(t),t)+diff(y(t),t) = 0, diff(diff(x(t),t),t)+diff(diff(y(t),t),t)-x(t) = 0];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) c_3 \,{\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) c_4 \,{\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}+{\mathrm e}^{t} c_1 \\
y \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+c_4 \,{\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.017 (sec). Leaf size: 246
ode={D[x[t],{t,2}]-D[x[t],t]+D[y[t],t]==0,D[x[t],{t,2}]+D[y[t],{t,2}]-x[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to -\frac {1}{10} e^{\frac {1}{2} \left (t-\sqrt {5} t\right )} \left (2 c_1 \left (\sqrt {5} e^{\sqrt {5} t}-5 e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}-\sqrt {5}\right )-2 \sqrt {5} c_2 \left (e^{\sqrt {5} t}-1\right )+c_4 \left (\left (5+\sqrt {5}\right ) e^{\sqrt {5} t}-10 e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}+5-\sqrt {5}\right )\right )\\ y(t)&\to \frac {1}{10} \left (\left (5+\sqrt {5}\right ) c_1-\left (5+\sqrt {5}\right ) c_2-2 \sqrt {5} c_4\right ) e^{\frac {1}{2} \left (t-\sqrt {5} t\right )}+\frac {1}{10} \left (-\left (\left (\sqrt {5}-5\right ) c_1\right )+\left (\sqrt {5}-5\right ) c_2+2 \sqrt {5} c_4\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}-c_1+c_2+c_3 \end{align*}
✓ Sympy. Time used: 0.157 (sec). Leaf size: 82
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-Derivative(x(t), t) + Derivative(x(t), (t, 2)) + Derivative(y(t), t),0),Eq(-x(t) + Derivative(x(t), (t, 2)) + Derivative(y(t), (t, 2)),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} e^{t} - C_{2} e^{\frac {t \left (1 - \sqrt {5}\right )}{2}} - C_{3} e^{\frac {t \left (1 + \sqrt {5}\right )}{2}}, \ y{\left (t \right )} = - \frac {C_{2} \left (1 + \sqrt {5}\right ) e^{\frac {t \left (1 - \sqrt {5}\right )}{2}}}{2} - \frac {C_{3} \left (1 - \sqrt {5}\right ) e^{\frac {t \left (1 + \sqrt {5}\right )}{2}}}{2} + C_{4}\right ]
\]