how to write to solve this type of system of equations ?
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Wan Ji on 12 Sep 2021
Just expand the left items of the two equations, extract u'' and v'', then an ode45 solver is there for you.
Walter Roberson on 12 Sep 2021
Edited: Walter Roberson on 12 Sep 2021
I was not able to figure out what is being raised to 10/9 . I used squiggle instead.
syms u(r) v(r)
syms N squiggle real
du = diff(u);
dv = diff(v);
left1 = diff(r^(N-1)*du^3)
right1 = r^(N-1) * sqrt(u) * sqrt(v) / (3*r^(2/3) * sqrt(1 + 9*squiggle^(10/9)/(10*(3*N-2)^(1/3))))
left2 = diff(r^(N-1)*dv^3)
right2 = r^(N-1) * u * v / (3*r^(2/3))
eqn1 = left1 == right1
eqn2 = left2 == right2
ic = [u(0) == 1, v(0) == 1, du(0) == 0, dv(0) == 0]
sol = dsolve([eqn1, eqn2, ic])
Lack of a symbolic solution means that you would have to do numeric solutions -- but you cannot do a numeric solution to infinity, and you certainly would not get a formula out of it.