60.9.37 problem 1892
Internal
problem
ID
[11891]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1892
Date
solved
:
Tuesday, January 28, 2025 at 06:24:02 PM
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*}
✓ Solution by Maple
Time used: 0.132 (sec). Leaf size: 867
dsolve([diff(x(t),t,t)-a*diff(y(t),t)+b*x(t)=0,diff(y(t),t,t)+a*diff(x(t),t)+b*y(t)=0],singsol=all)
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_3 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \\
y \left (t \right ) &= \frac {c_{1} \left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{{3}/{2}} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{1} a^{2}-c_{2} \left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{{3}/{2}} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{2} a^{2}+c_3 \left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{{3}/{2}} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_3 \,a^{2}-c_4 \left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{{3}/{2}} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_4 \,a^{2}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{1} b -4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{2} b +4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_3 b -4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_4 b}{8 a b} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.160 (sec). Leaf size: 3522
DSolve[{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},{x[t],y[t]},t,IncludeSingularSolutions -> True]
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