Some ideas for particular solutions? How to make two Wronskian?
a1*x''''[t] + b1*x''[t] + c1*y''''[t] + d1*y''[t] + e1*y[t] == f1[t],
a2*y''''[t] + b2*y''[t] + c2*x''''[t] + d2*x''[t] + e2*x[t] == f2[t],
All constants are known: a1, b1,...
Some ideas for particular solutions? How to make two Wronskian?
a1*x''''[t] + b1*x''[t] + c1*y''''[t] + d1*y''[t] + e1*y[t] == f1[t],
a2*y''''[t] + b2*y''[t] + c2*x''''[t] + d2*x''[t] + e2*x[t] == f2[t],
All constants are known: a1, b1,...
In first line it should be (-a2/c2)*y'''' then we will get y4 = ((-a1 b2 + c2 d1) y'')/(a1 a2 - c1 c2) + (c2 e1 y)/(a1 a2 - c1 c2) + ((b1 c2 - a1 d2) x'')/(a1 a2 - c1 c2) - (a1 e2 x)/(a1 a2 -c1 c2) + (a1 f2)/(a1 a2 - c1 c2). How further? We can use substitutions and we must get 8 equations?
These are the 8 equations
and
I will do a bit of the x's
1st equation
2nd equation
The last step used equation 1
3rd equation
the forth x equations comes from the algebra that I did in the first post
This is the first 4 equations, Now find the other 4 y equations in the exact same way then write the system as
In fourth equation when we use the substitution (e2/c2)*x1 where x1=dx/dt. x1 is not correct because it should be (e2/c2)*x in equation? What I should to put instead of x or y to make for equation? I understand that in fourth equation figure x'', y'',x and y. For x''=x2, for y''=y, but what to put for x and y without differentiation?
Btw I got x'''' and y'''' after some algebra from first equations in the following form:
x'''' = (a2 f1)/(a1 a2 - c1 c2) - (c1 f2)/(a1 a2 - c1 c2) + ((b2 c1 - a2 e1)/(a1 a2 - c1 c2)) y - ((a2 d1 )/(a1 a2 - c1 c2)) y'' + ((-a2 b1 + c1 e2)/(a1 a2 - c1 c2)) x + ((c1 d2 )/(a1 a2 - c1c2)) x''
y'''' = (c2 f1)/(-a1 a2 + c1 c2) - (a1 f2)/(-a1 a2 +c1 c2) + ((-a1 b2 + c2 e1 )/(a1 a2 - c1 c2)) y -(c2 d1 )/(-a1 a2 + c1 c2)y'' + (((b1 c2 - a1 e2) )/(a1 a2 - c1 c2)) x + ((a1 d2)/(-a1 a2 + c1 c2)) x''
And after that what to put in fourth and 8th diff equation (first order) instead x and y? I understand what to put instead x'' and y''. But what for x and y?
Yes You are correct that there is a typo in the equaiton. It should read
You don't need to put anything they stay in the equations.
Your variables in the columns will look like
The matrix will have 8 rows and 8 columns and only the 4th and 8th rows will have more than one entry
As above this will give
We didn't get the same equations after algebra. But it is not a problem. I got following:
x'''' = (a2 f1)/(a1 a2 - c1 c2) - (c1 f2)/(a1 a2 - c1 c2) + ((b2 c1 - a2 e1)/(a1 a2 - c1 c2)) y - ((a2 d1 )/(a1 a2 - c1 c2)) y'' + ((-a2 b1 + c1 e2)/(a1 a2 - c1 c2)) x + ((c1 d2 )/(a1 a2 - c1c2)) x''
y'''' = (c2 f1)/(-a1 a2 + c1 c2) - (a1 f2)/(-a1 a2 +c1 c2) + ((-a1 b2 + c2 e1 )/(a1 a2 - c1 c2)) y -(c2 d1 )/(-a1 a2 + c1 c2)y'' + (((b1 c2 - a1 e2) )/(a1 a2 - c1 c2)) x + ((a1 d2)/(-a1 a2 + c1 c2)) x''
I will try to solve with this equations. You help me a lot anyway. If it is not a problem, write me the complete form of matrix equation. X'=AX+F(t). In other problem with variation constants we use wronksian. I am afraid to make a mistake in matrix equation. Thank you anyway!
Thank you for matrix equation in complete form. That it is understandable. But from initial equations I solve the equations where x'''' and y'''' are unknown, also put in Mathematica software and got x'''' in function of (x'',x) and y'''' in function of (y'',y). Mr Set, you got from second equation x'''' where I can see (a2+b2)/c2 , but a2 multiplied y'''', and I can't see y''''. Anyway thank you a lot for matrix and everything.