Given the Volterra integral equation
$$y(x)=x-\int_0^x xt^2 y(t)\,dt$$
How do I solve it? Predicting $y$ using iteration is seeming difficult. Please help.
$\endgroup$ 21 Answer
$\begingroup$We start from this equation: $$y(x) = x\left(1 - \int_0^x t^2 y(t) dt \right)$$
Differentiating both sides, we get:
$$\frac{dy(x)}{dx} = \left(1 - \int_0^x t^2 y(t) dt \right) - x^3y(x).$$
Notice that:
$$y(x) = x\left(1 - \int_0^x t^2 y(t) dt \right) \Rightarrow \left(1 - \int_0^x t^2 y(t) dt \right) = \frac{y(x)}{x}.$$
Then: $$\frac{dy(x)}{dx} = \frac{y(x)}{x} - x^3y(x) \Rightarrow \\y' = y\left(\frac{1}{x} - x^3\right).$$
Now, we can separate the variables:
$$\int \frac{dy}{y} = \int \left(\frac{1}{x} - x^3\right)dx \Rightarrow \\ \log(y) = \log(x) - \frac{x^4}{4} + C \Rightarrow \\ y = e^{\log(x) - \frac{x^4}{4} + C} \Rightarrow\\ y = Ax e^{-\frac{x^4}{4}}.$$
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