how to solve ode which includes ode and integral

How can I solve the following equations:

df(x,y)/dx= y*f^2(x,y)-dg(x)/dx*f(x,y)/g(x)+y*g(x)

dg(x)/dx=2g(x)*integral{f(x,y)dy} ... (y goes from -t to +t)

with given f(0,y), and g(0).

[RahulTandon]

THIS IS A ODE ALRIGHT, BUT A SYMBOLIC MATH PROBLEM. IT IS NOT AN ODE45 PROBLEM! OK BY YOU! BELOW IS THE SOLUTION!

[RahulTandon]

i think the function f(x,y) will have to be futher specified to get a particular solution to the proble!

Answer 1

[RahulTandon]

clc; syms t x y f(x,y) g(x) k1 k2; Sol = dsolve('Dg - 2*g*int(f,y)','Dx - y*f^2- f/g*Dg','g(0)==k1','IgnoreAnalyticConstraints', true,'y'); % we obtain the symbolic solution to the problem Sol2 = solve(Sol.x==0,Sol.g==0,f(0,y)==k2); Sol.x

Answer 2

[Torsten]

Choose an equidistant grid in y-direction: yi=-t+(i-1)/(n-1)*2t (i=1,...,n).

Call the ODE Integrator (ODE15s,e.g.) for the n+1 differential equations

df(x,yi)/dx= yi*f^2(x,yi)-2g(x)*integral{f(x,y)dy}*f(x,yi)/g(x)+yi*g(x)
dg(x)/dx=2g(x)*integral{f(x,y)dy}

and approximate the integral via the trapezoidal rule:

integral_{y=-t}^{y=t}{f(x,y)dy}=dy/2*sum_{i=1}^{i=n-1}(f(x,yi)+f(x,y(i+1)))

with dy = 2t/(n-1)

Best wishes

Torsten.