54.9.10 problem 1865
Internal
problem
ID
[13088]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1865
Date
solved
:
Wednesday, October 01, 2025 at 03:34:23 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}\\ \frac {d}{d t}y \left (t \right )&=a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2} \end{align*}
✓ Maple. Time used: 0.162 (sec). Leaf size: 333
ode:={diff(x(t),t) = a__1*x(t)+b__1*y(t)+c__1, diff(y(t),t) = a__2*x(t)+b__2*y(t)+c__2};
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (\frac {a_{1}}{2}+\frac {b_{2}}{2}+\frac {\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}{2}\right ) t} c_2 +{\mathrm e}^{\left (\frac {a_{1}}{2}+\frac {b_{2}}{2}-\frac {\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}{2}\right ) t} c_1 +\frac {c_{2} b_{1} -b_{2} c_{1}}{a_{1} b_{2} -a_{2} b_{1}} \\
y \left (t \right ) &= \frac {-\frac {a_{1} \left ({\mathrm e}^{\frac {\left (a_{1} +b_{2} +\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_2 \left (a_{1} b_{2} -a_{2} b_{1} \right )+{\mathrm e}^{\frac {\left (a_{1} +b_{2} -\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_1 \left (a_{1} b_{2} -a_{2} b_{1} \right )+c_{2} b_{1} -b_{2} c_{1} \right ) \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{a_{1} b_{2} -a_{2} b_{1}}+\frac {\left (a_{1} +b_{2} +\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) {\mathrm e}^{\frac {\left (a_{1} +b_{2} +\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_2 \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{2}+\frac {\left (a_{1} +b_{2} -\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) {\mathrm e}^{\frac {\left (a_{1} +b_{2} -\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_1 \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{2}-c_{1} \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{\left (2 a_{1} b_{2} -2 a_{2} b_{1} \right ) b_{1}} \\
\end{align*}
✓ Mathematica. Time used: 0.743 (sec). Leaf size: 1684
ode={D[x[t],t]==a1*x[t]+b1*y[t]+c1,D[y[t],t]==a2*x[t]+b2*y[t]+c2};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 0.695 (sec). Leaf size: 1574
from sympy import *
t = symbols("t")
a__1 = symbols("a__1")
a__2 = symbols("a__2")
b__1 = symbols("b__1")
b__2 = symbols("b__2")
c__1 = symbols("c__1")
c__2 = symbols("c__2")
x = Function("x")
y = Function("y")
ode=[Eq(-a__1*x(t) - b__1*y(t) - c__1 + Derivative(x(t), t),0),Eq(-a__2*x(t) - b__2*y(t) - c__2 + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\text {Solution too large to show}
\]