60.9.39 problem 1894
Internal
problem
ID
[11893]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1894
Date
solved
:
Tuesday, January 28, 2025 at 06:24:02 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+a \left (\frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )\right )+b_{1} x \left (t \right )&=c_{1} {\mathrm e}^{i \omega t}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+a \left (\frac {d}{d t}y \left (t \right )-\frac {d}{d t}x \left (t \right )\right )+b_{2} y \left (t \right )&=c_{2} {\mathrm e}^{i \omega t} \end{align*}
✓ Solution by Maple
Time used: 0.796 (sec). Leaf size: 2459
dsolve([diff(x(t),t,t)+a*(diff(x(t),t)-diff(y(t),t))+b__1*x(t)=c__1*exp(I*omega*t),diff(y(t),t,t)+a*(diff(y(t),t)-diff(x(t),t))+b__2*y(t)=c__2*exp(I*omega*t)],singsol=all)
\begin{align*}
x \left (t \right ) &= \frac {i {\mathrm e}^{i \omega t} c_{1} a \omega +i {\mathrm e}^{i \omega t} c_{2} a \omega -{\mathrm e}^{i \omega t} \omega ^{2} c_{1} +{\mathrm e}^{i \omega t} c_{1} b_{2}}{-2 i a \,\omega ^{3}+i a b_{1} \omega +i a b_{2} \omega +\omega ^{4}-\omega ^{2} b_{1} -b_{2} \omega ^{2}+b_{1} b_{2}}+c_3 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (a b_{1} +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} =1\right ) t}+c_4 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (a b_{1} +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} =2\right ) t}+c_5 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (a b_{1} +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} =3\right ) t}+c_6 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{3}+\left (b_{1} +b_{2} \right ) \textit {\_Z}^{2}+\left (a b_{1} +b_{2} a \right ) \textit {\_Z} +b_{1} b_{2} , \operatorname {index} =4\right ) t} \\
\text {Expression too large to display} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.131 (sec). Leaf size: 3386
DSolve[{D[x[t],{t,2}]+a*(D[x[t],t]-D[y[t],t])+b1*x[t]==c1*Exp[I*\[Omega]*t],D[y[t],{t,2}]+a*(D[y[t],t]-D[x[t],t])+b2*y[t]==c2*Exp[I*\[Omega]*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
Too large to display