60.9.32 problem 1887
Internal
problem
ID
[11811]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1887
Date
solved
:
Sunday, March 30, 2025 at 09:15:51 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*}
✓ Maple. Time used: 0.165 (sec). Leaf size: 417
ode:=[diff(diff(x(t),t),t) = a*x(t)+b*y(t), diff(diff(y(t),t),t) = c*x(t)+d*y(t)];
dsolve(ode);
\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*}
✓ Mathematica. Time used: 0.22 (sec). Leaf size: 5647
ode={D[x[t],{t,2}]==a*x[t]+b*y[t],D[y[t],{t,2}]==c*x[t]+d*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 2.829 (sec). Leaf size: 717
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
x = Function("x")
y = Function("y")
ode=[Eq(-a*x(t) - b*y(t) + Derivative(x(t), (t, 2)),0),Eq(-c*x(t) - d*y(t) + Derivative(y(t), (t, 2)),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {\sqrt {2} C_{1} b \sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} e^{- \frac {\sqrt {2} t \sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{a d - 2 b c - d^{2} + d \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} - \frac {\sqrt {2} C_{2} b \sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} e^{\frac {\sqrt {2} t \sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{a d - 2 b c - d^{2} + d \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} + \frac {\sqrt {2} C_{3} b \sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} e^{\frac {\sqrt {2} t \sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{- a d + 2 b c + d^{2} + d \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} - \frac {\sqrt {2} C_{4} b \sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} e^{- \frac {\sqrt {2} t \sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{- a d + 2 b c + d^{2} + d \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}, \ y{\left (t \right )} = - \frac {\sqrt {2} C_{1} e^{- \frac {\sqrt {2} t \sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{\sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}} + \frac {\sqrt {2} C_{2} e^{\frac {\sqrt {2} t \sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{\sqrt {a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}} + \frac {\sqrt {2} C_{3} e^{\frac {\sqrt {2} t \sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{\sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}} - \frac {\sqrt {2} C_{4} e^{- \frac {\sqrt {2} t \sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}{2}}}{\sqrt {a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}}\right ]
\]