Euler method. Runge-Kuta methods. Separable first-order ODEs.

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 โˆ†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.

Linear first-order ODEs.
Application: compound interest. Application: terminal velocity.
Application: RC Circuit.

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