Separating the variables to solve differential equations is a familiar and simple method, but limited in it’s usefulness because most equations are not separable. If however the equation is of the form whereandare both of the formthen separability can be achieved with the substitution

Proof: ifthen

Simplification of the right hand side returns

Now x cancels throughout to giveand this equation is separable.

Example Use the substitutionto transform and solve the differential equationand solve it subject toat

Separating the variables gives

Now we can integrate:

when

Multiply byto get