We now come to a very important class of DEs : first-order linear DEs, their importance arising from the fact that many natural phenomena can be described using such DEs.
First order linear DEs take the form
where and are functions of alone.
To solve such DEs, the method followed is as described below :
We multiply both sides of the DE by a quantity called the integrating factor (I.F.) where
Why this is chosen as the I.F. will soon become clear when we see what the I. F. actually does :
The left hand side now becomes exact, in the sense that it can be expressed as the exact differential of some expression :
Now our DE becomes
This can now easily be integrated to yield the required general solution:
You are urged to re-read this discussion until you fully understand its significance. In particular, you must understand why multiplying the DE by the I. F. on both sides reduces its left hand side to an exact differential.
Solve the DE
Comparing this DE with the standard form of the linear DE , we see that
Thus, the I.F. is
Multiplying by on both sides of the given DE, we obtain
The left hand side is an exact differential :
Integrating both sides, we obtain the solution to our DE as