60.9.5 problem 1860
Internal
problem
ID
[11784]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1860
Date
solved
:
Sunday, March 30, 2025 at 09:15:14 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=a x \left (t \right )+b y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=c x \left (t \right )+b y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.127 (sec). Leaf size: 236
ode:=[diff(x(t),t) = a*x(t)+b*y(t), diff(y(t),t) = c*x(t)+b*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (a +b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} \\
y \left (t \right ) &= \left (\frac {1}{2}+\frac {\frac {\sqrt {a^{2}-2 a b +b^{2}+4 b c}}{2}-\frac {a}{2}}{b}\right ) c_1 \,{\mathrm e}^{\frac {\left (a +b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}+\left (\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}}{2}+\frac {-\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} \sqrt {a^{2}-2 a b +b^{2}+4 b c}}{2}-\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} a}{2}}{b}\right ) c_2 \\
\end{align*}
✓ Mathematica. Time used: 0.028 (sec). Leaf size: 362
ode={D[x[t],t]==a*x[t]+b*y[t],D[y[t],t]==c*x[t]+b*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )} \left (a c_1 \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )+c_1 \sqrt {a^2-2 a b+b^2+4 b c} \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}+1\right )-b (c_1-2 c_2) \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )\right )}{2 \sqrt {a^2-2 a b+b (b+4 c)}} \\
y(t)\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )} \left (2 c c_1 \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )+c_2 \left (a \left (-e^{t \sqrt {a^2-2 a b+b^2+4 b c}}\right )+b \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )+\sqrt {a^2-2 a b+b^2+4 b c} \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}+1\right )+a\right )\right )}{2 \sqrt {a^2-2 a b+b (b+4 c)}} \\
\end{align*}
✓ Sympy. Time used: 0.304 (sec). Leaf size: 180
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
x = Function("x")
y = Function("y")
ode=[Eq(-a*x(t) - b*y(t) + Derivative(x(t), t),0),Eq(-b*y(t) - c*x(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {2 C_{1} b e^{\frac {t \left (a + b + \sqrt {a^{2} - 2 a b + b^{2} + 4 b c}\right )}{2}}}{- a + b + \sqrt {a^{2} - 2 a b + b^{2} + 4 b c}} - \frac {2 C_{2} b e^{\frac {t \left (a + b - \sqrt {a^{2} - 2 a b + b^{2} + 4 b c}\right )}{2}}}{a - b + \sqrt {a^{2} - 2 a b + b^{2} + 4 b c}}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (a + b + \sqrt {a^{2} - 2 a b + b^{2} + 4 b c}\right )}{2}} + C_{2} e^{\frac {t \left (a + b - \sqrt {a^{2} - 2 a b + b^{2} + 4 b c}\right )}{2}}\right ]
\]