Question

Solve the separable differential equation for u.

Answer 1

The solution to the separable differential equation du/dx = f(x)g(u) with the initial condition
`u(x_0) = u_0` is given by
`u(x) = [∫ g^(-1)(u_0) du] / [∫ g^(-1)(u) du]` , where
`g^(-1)` represents the inverse function of g(u).

Step-by-step explanation:

The separable differential equation du/dx = f(x)g(u) can be solved by separating the variables involving u and x, followed by integrating both sides. After separation, you'll typically have terms involving u and du on one side and terms involving x and dx on the other.

Integrating each side separately will give an equation relating u and x. Then, applying the initial condition
`u(x_0) = u_0` allows you to solve for u(x).

The solution involves integrating the inverse function of g(u) with respect to u, which can be represented as
`g^(-1)(u)`, allowing you to solve for u(x) by using definite integrals and the initial condition.

Here is complete question;

"Solve the separable differential equation du/dx = f(x)g(u) for u(x), given the initial condition
`u(x_0) = u_0`."