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 haveandterms on both sides, but if the equation is of a certain form –- we can rearrange to have all terms includingon the right hand side and all terms includingon the left hand side, obtaining in this case,

We can then integrate both sides:

Example: Solve the differential equation

Multiply byand divide byto giveWe can now integrate:

is a product which we integrate by parts obtainingTo find the constantwe need what is called a boundary condition. Suppose then that we have that whenSubstitute these values into (1) to obtain


Example: Solve

Factorise the right hand side intoto givewhich is separable.

Multiply byand divide byto giveWe can now integrate:(1)

If we are to makethe subject we exponentiate both sides, raisingto the power of both sides:

whereNotice how the constant termin (1) becomes the constant factorwhen we exponentiate.