Differential Equations - Separation of Variables

Solving a differential equation by separation of variables is usually the first thing that shows up in an introduction to differential equations. The basic idea is just to group the variables on one side or another, as in the following example. The main tricky part is just to remember the different ways in which a partial derivative can be written.

y'=dy/dx=y_x

Are all equivalent. Also, although its usually preferred to be written in a more descriptive way, d/dx is sometimes written as D. Now lets work through an example.

y' = -4x/y

y dy/dx = -4x

y dy + 4x dx = 0

y²/2 + 2x² = C

y² + 4x² = C

The reason I moved the 4x over to the left hand side is because since we know that the two will integrate to x^2 and y^2, the solution must be an ellipse with the size defined as the combination of the constants due to integration left on the right hand side. Thus if we have some initial values, such as

y(2)=3

Thus our constant will solve out to

C = 3² + 4(2)²

Giving us a solution of

C = 25
y² + 4x² = 25

Although basic there is a lot of use to separation of variables. Combining separation of variables and reasoning through the specified class of functions we know the system must behave like is taught as proofs for many of the equations in electrostatics.