# First-Order Differential Equations ## Euler Method

Euler method is a very simple numerical method that can be used to solve differential equations. For example `dy / dx = f(x, y)` , there is usually no such a thing as an analytical solution, so one has to solve numerically with an initial condition `y(x0) = y0` . Numerical methods approximate the solution `y(x)` as a line segment that follows the slope `dy / dx` , getting new point `x1, y1` . Then the new point becomes the initial condition, we repeat the process.

So if the ∆x is small enough, you will have all these lines segments all glued together, it will look like a curve, which is the solution. Occasionally this can go wrong, in case that f(x, y) is not well-behaving function, say infinity, imaginary, etc.

## Separable First-Order ODEs

``g(y) dy/dx = f(x)``

If a first-order differential equation `dy/dx` can be expressed as a function of x multiplied by a function of y, then we say it is separable, where it is possible to find an analytical solution. A separable equation can be integrated, and end up with a solution that you have an integral on the left, and an integral on the right. In many cases you can solve this equation for y as a function of x.

## Linear First-Order ODEs

``dy/dx + p(x) y = g(x)``

To solve it, the idea is multiplying it by something called integrating factor, which is going to help us to integrate the differential equation. Applications includes: compound interest, terminal velocity, RC circuit.

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