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


Substituteso that

The equation becomes

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

Cancelling ux from both sides gives


Integrating gives


We can find c using the boundary condition


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