60.9.38 problem 1893
Internal
problem
ID
[11817]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1893
Date
solved
:
Sunday, March 30, 2025 at 09:15:57 PM
CAS
classification
:
system_of_ODEs
\begin{align*} a_{1} \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+b_{1} \left (\frac {d}{d t}x \left (t \right )\right )+c_{1} x \left (t \right )-A \left (\frac {d}{d t}y \left (t \right )\right )&=B \,{\mathrm e}^{i \omega t}\\ a_{2} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+b_{2} \left (\frac {d}{d t}y \left (t \right )\right )+c_{2} y \left (t \right )+A \left (\frac {d}{d t}x \left (t \right )\right )&=0 \end{align*}
✓ Maple. Time used: 0.855 (sec). Leaf size: 1542
ode:=[a__1*diff(diff(x(t),t),t)+b__1*diff(x(t),t)+c__1*x(t)-A*diff(y(t),t) = B*exp(I*omega*t), a__2*diff(diff(y(t),t),t)+b__2*diff(y(t),t)+c__2*y(t)+A*diff(x(t),t) = 0];
dsolve(ode);
\begin{align*}
\text {Expression too large to display} \\
y \left (t \right ) &= \frac {i B \,{\mathrm e}^{i \omega t} \omega A}{-a_{1} a_{2} \omega ^{4}+i a_{1} b_{2} \omega ^{3}+i a_{2} b_{1} \omega ^{3}+A^{2} \omega ^{2}+a_{1} c_{2} \omega ^{2}+a_{2} c_{1} \omega ^{2}+b_{1} b_{2} \omega ^{2}-i b_{1} c_{2} \omega -i b_{2} c_{1} \omega -c_{1} c_{2}}+c_3 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+a_{1} c_{2} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{1} c_{2} , \operatorname {index} =1\right ) t}+c_4 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+a_{1} c_{2} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{1} c_{2} , \operatorname {index} =2\right ) t}+c_5 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+a_{1} c_{2} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{1} c_{2} , \operatorname {index} =3\right ) t}+c_6 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+a_{1} c_{2} +a_{2} c_{1} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{1} c_{2} , \operatorname {index} =4\right ) t} \\
\end{align*}
✓ Mathematica. Time used: 0.204 (sec). Leaf size: 5546
ode={a1*D[x[t],{t,2}]+b1*D[x[t],t]+c1*x[t]-A*D[y[t],t]==B*Exp[I*\[Omega]*t],a2*D[y[t],{t,2}]+b2*D[y[t],t]+c2*y[t]+A*D[x[t],t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Too large to display
✗ Sympy
from sympy import *
t = symbols("t")
A = symbols("A")
B = symbols("B")
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")
omega = symbols("omega")
x = Function("x")
y = Function("y")
ode=[Eq(-A*Derivative(y(t), t) - B*exp(omega*t*complex(0, 1)) + a__1*Derivative(x(t), (t, 2)) + b__1*Derivative(x(t), t) + c__1*x(t),0),Eq(A*Derivative(x(t), t) + a__2*Derivative(y(t), (t, 2)) + b__2*Derivative(y(t), t) + c__2*y(t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
Timed Out