Given the integral equation: $\sqrt{f(x)}\int_{0}^{x}f(\tau)d\tau=g(x)$ with g(x) known function, in what cases and how is it possible to solve it?
Thanks
Given the integral equation: $\sqrt{f(x)}\int_{0}^{x}f(\tau)d\tau=g(x)$ with g(x) known function, in what cases and how is it possible to solve it?
Thanks
Square both sides of the equation : $F^'(x) \ (F(x))^2=(g(x))^2$ with $F(x) = \int_{0}^{x}f(\tau)d\tau $.
Now by integration between 0 and t : $\frac{(F(t))^3}{3} = \int_{0}^{t} (g(x))^2 dx$ F(t) can now be expressed : $F(t) =\sqrt[3]{3} \ \left( \int_{0}^{t} (g(x))^2 dx \right)^{1/3}$ since $F'(t)=f(t)$, by derivation we obtain : $f(t) = \sqrt[3]{3} \ (g(t))^2 \left(\int_{0}^{t} (g(x))^2 dx \right)^{-2/3}$
Let us assume that $g(x)$ is differentiable (in fact $g(x)$ should be also positive and monotonously increasing $f(x)$ has to be positive such that the square root is properly defined).
Take a derivative of your equation and you obtain $ f(x)^{3/2} + \frac{g(x)\, f'(x)}{2 f(x)} =g'(x). $ Thus, you have reduced your equation to a differential equation with the initial condition $f(0) = g(0)$.