A Differential equation with integral

[petterson]

Hi,
I was wondering if anyone had some advice on how to solve the following equation for $F(b)$:
$$ F(g(b)) h(b) + F'(b) s(b) - F(b)h(b) + h(b) + \int_{g(b)}^{b} v(x) F'(x) dx = 0 $$

Any hints on how to tackle this would be highly appreciated. Thank you!

 

[andrewkirk]

If it's the presence of the integral that throws you off, you can get rid of it by differentiating both sides with respect to b. For the part that is the integral, we have

$$
\frac d{db}\left(\int_{g(b)}^{b} v(x) F'(x) dx\right)
=v(b) F'(b)-
\left(v(x) F'(x))
\right)g'(b)
$$
To see this, let the function $A$ be the antiderivative of the integrand. Then the definite integral is
$$A(b)-A(g(b))$$
We differentiate that wrt b and use the fact that $\frac {dA(y)}{dy}=v(y) F'(y)$ to get the above formula.

 

[petterson]

Thanks andrewkirk. Doing what you proposed the equation looks nicer, I obtain the following:

$$ F'(g(b)) g'(b) [h(b) - v(g(b))] + F(g(b)) h'(b) + F''(b)s(b) + F'(b) [s'(b) + v(b) - h(b)] - F(b) h'(b) + h'(b) = 0 $$

There is still the issue with those nested function $F(g(b))$ and $F'(g(b))$. How could I deal with that?

 

[petterson]

I was able to simplify my problem to the following equation:
$$ F(g(b)) - F(b) + F'(b)h(b) + F''(b) s(b) - k = 0 $$
$g(b)$ is a decreasing function, k a constant. If necessary, one could also assume that $g(b)$ is an affine function. Does the delay differential equation framework apply here? I'm not sure since g(b) is decreasing. Could one possibly get a numerical solution with the ddesd function in Matlab?
Thanks!