60.9.43 problem 1898
Internal
problem
ID
[11897]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1898
Date
solved
:
Tuesday, January 28, 2025 at 06:24:04 PM
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*}
✓ Solution by Maple
Time used: 0.100 (sec). Leaf size: 72
dsolve([diff(x(t),t,t)-diff(x(t),t)+diff(y(t),t)=0,diff(x(t),t,t)+diff(y(t),t,t)-x(t)=0],singsol=all)
\begin{align*}
x \left (t \right ) &= \left (-\frac {1}{2}-\frac {\sqrt {5}}{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}}+c_{1} {\mathrm e}^{t} \\
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*}
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 246
DSolve[{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},{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*}