## Differential Equations - Separating Variables

It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables – the technique illustrated here. In general a differential equation may have and terms on both sides, but if the equation is of a certain form – - we can rearrange to have all terms including on the right hand side and all terms including on the left hand side, obtaining in this case, We can then integrate both sides: Example: Solve the differential equation Multiply by and divide by to give We can now integrate:  is a product which we integrate by parts obtaining To find the constant we need what is called a boundary condition. Suppose then that we have that when Substitute these values into (1) to obtain hence Example: Solve Factorise the right hand side into to give which is separable.

Multiply by and divide by to give We can now integrate: (1)

If we are to make the subject we exponentiate both sides, raising to the power of both sides: where Notice how the constant term in (1) becomes the constant factor when we exponentiate.

#### Add comment Refresh