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
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:
Multiply byto get