Solving Differential Equations By Separation of Variables

Some differential equations are difficult to solve as presented but can be made much simpler using a suitable transformation of variables in the same way that an integral can be made much simpler by a suitable substitution. A very simple example is

(when)

Substituteso that

The equation becomes

Exapanding the brackets gives x^2 {du over dx} +ux =x +ux

Cancelling ux from both sides gives

Hence

Integrating gives

Sinceso

We can find c using the boundary condition

Hence