8.2.1. 1st Order LDE

A linear differential equation (LDE) of 1st order takes the form

        y +P( x )y=Q( x )                  ( y = dy dx )         (8.1)

where P and Q are known functions and y is the unknown.  This is one of the few type of DEs that can be solved by elementary means, i.e., through algebraic operations, compositions, and integrations.

Theorem 8.1.

Let P and Q be continuous on an open interval J.  Then there exists 1 and only 1 function y=f( x )  that satisfies both (8.1) and the initial condition f( a )=b , where aJ  and bR Furthermore,

        f( x )=b e A( x ) + e A( x ) a x dt   Q( t ) e A( t )                         (8.2)

where  A( x )= a x dt   P( t ) .

Proof

Using A ( x )=P( x ) , we see that e A( x ) ×( 8.1 )  gives

        e A y + e A A y= e A Q  

Þ                  ( e A y ) = e A Q                    (8.1a)

The factor e A( x )  is called an integrating factor since we can now integrate (8.1a) over the interval [ a,x ]  to get

        e A( x ) f( x ) e A( a ) f( a )= a x dt   Q( t ) e A( t )

Using A( a )= a a dt   P( t )=0  and f( a )=b , eq(8.2) is recovered.  Uniqueness of this solution is guaranteed by Theorem 8.3 of §8.5.