60.9.32 problem 1887
Internal
problem
ID
[11886]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1887
Date
solved
:
Tuesday, January 28, 2025 at 06:24:00 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=a x \left (t \right )+b y \left (t \right )\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=c x \left (t \right )+d y \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.128 (sec). Leaf size: 417
dsolve([diff(x(t),t,t)=a*x(t)+b*y(t),diff(y(t),t,t)=c*x(t)+d*y(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_3 \,{\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_4 \,{\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}} \\
y \left (t \right ) &= \left (\frac {d}{2 b}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}-\frac {a}{2}}{b}\right ) c_4 \,{\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (\frac {d}{2 b}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}-\frac {a}{2}}{b}\right ) c_3 \,{\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (\frac {d}{2 b}+\frac {-\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}-\frac {a}{2}}{b}\right ) c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (\frac {d}{2 b}+\frac {-\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}-\frac {a}{2}}{b}\right ) c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.220 (sec). Leaf size: 5647
DSolve[{D[x[t],{t,2}]==a*x[t]+b*y[t],D[y[t],{t,2}]==c*x[t]+d*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
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