How can i solve a this complex differential equation using ode45?

Good morning.

For my training, i would like to solve this differential equation:

     (K1*y')^2 + [K1+(K1^2/K2)]*y' - 1 =0

where K1 = sin(y)/[(1+cos(y))*(2-cos(y))]

      K2 = 0.31498*cos(y)*sin(y)/[((1+cos(y))^4/3)*((2-cos(y))^5/3)]

I have already tried to solve it with ode45, but as you see, K1 and K2 are funtions of y and the power 2 complicates the equation.

I have 2 weeks to finish it. Please help me by giving me advices.

Thank you

DIALLO

Answer 1

[Star Strider]

Interesting problem. I suggest you start by considering it a quadratic expression of y', then solving it as a system of two differential equations in y'. Starting with the Symbolic Math Toolbox:

syms K1 K2 dy
yprime = solve( (K1*dy)^2 + [K1+(K1^2/K2)]*dy - 1 == 0, dy )

yields:

yprime =

   -(K1 + K2 + (K1^2 + 2*K1*K2 + 5*K2^2)^(1/2))/(2*K1*K2)
   -(K1 + K2 - (K1^2 + 2*K1*K2 + 5*K2^2)^(1/2))/(2*K1*K2)

Now you have a system of two first-order differential equations. When I solved it with ode45 with initial conditions of [1 1], I got two curves asymptotic at about ±1.57.