60.9.39 problem 1894

Internal problem ID [11818]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1894
Date solved : Friday, March 14, 2025 at 02:58:29 AM
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*}

✓ Maple. Time used: 0.796 (sec). Leaf size: 2459

ode:=[diff(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(diff(y(t),t),t)+a*(diff(y(t),t)-diff(x(t),t))+b__2*y(t) = c__2*exp(I*omega*t)]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 3386

ode={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]}; 
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__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(x(t), t) - Derivative(y(t), t)) + b__1*x(t) - c__1*exp(omega*t*complex(0, 1)) + Derivative(x(t), (t, 2)),0),Eq(a*(-Derivative(x(t), t) + Derivative(y(t), t)) + b__2*y(t) - c__2*exp(omega*t*complex(0, 1)) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Timed Out