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Could s.o. help me with Verifying this system of differential equation

asked 2022-11-21 14:57:02 +0100

Kaius gravatar image

updated 2022-11-23 08:37:43 +0100

vdelecroix gravatar image

I would like to verify that these functions :

F1=(sqrt(3)*sinh(sqrt(3)*x)+3*sin(x))/(3*cos(x)+4-cosh(sqrt(3)*x))
F2=(2*sqrt(3)*sin((1/2)*x)*sinh((1/2)*sqrt(3)*x)+6*cos((1/2)*x)*cosh((1/2)*sqrt(3)*x))/(3*cos(x)+4-cosh(sqrt(3)*x))
G1=(4*sqrt(3)*cos((1/2)*x)*sinh((1/2)*sqrt(3)*x))/(3*cos(x)+4-cosh(sqrt(3)*x))
G2=(6*cos((1/2)*x)*cosh((1/2)*sqrt(3)*x)+2*sqrt(3)*sin((1/2)*x)*sinh((1/2)*sqrt(3)*x)+2-2*cos(sqrt(3)*x))/(3*cos(x)+4-cosh(sqrt(3)*x))

are a solution to the system of differential equations:

diff(F1) == F2^2
diff(F2) == F1*F2+G1
diff(G1) == F2*(F2-G2)+F2
diff(G2) == F2*(F1-G1)+G1

Now, i have tried to solve the system directly with "desolve_system" which obviously didn't work. After that i just differentiated the functions above using sage and hoped that sage could somehow verify that e.g F1'=F2^2 but sadly i couldn't show that this was equal because they had such different forms that sage didn't recognize them as equal. But i know for a fact that these functions satisfy the system of DE. is there any way i could prove this using sage, and if so: how could i do it? and i am looking for an algebraic solution not a numeric one.

I would appreciate any answert to my post

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answered 2022-11-23 08:43:29 +0100

vdelecroix gravatar image

To check the first one

sage: bool(diff(F1) == F2^2)
True

However, I doubt the others are correct as evaluating at a specific x the difference you do not get a zero

sage: B = diff(F2) - (F1*F2 + G1)
sage: B.subs(x=0)
1
sage: C = diff(G1) - (F2*(F2-G2)+F2)
sage: C.subs(x=1).n()
10.5654556363032
sage: D = diff(G2) - (F2*(F1-G1)+G1)
sage: D.subs(x=0)
1
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Asked: 2022-11-21 14:56:22 +0100

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Last updated: Nov 23