I have the following exepression in my book: d x </mrow>

Dayanara Terry 2022-07-05 Answered
I have the following exepression in my book:
d x d t + a 1 ( t ) x = g ( t ) ,         x ( t 0 ) = x 0
Then it says, multiply both sides of the differential equation by the integrating factor I ( t ).
I ( t ) d x ( t ) d t + a 1 ( t ) I ( t ) x ( t ) = I ( t ) g ( t )
So far so good. Hereafter it says, the left-hand side is an exact derivative.
d [ x ( t ) I ( t ) ] d t = I ( t ) g ( t )
And my question is, how does the book come to the last? Can anyone give a HINT.
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Answers (1)

lywiau63
Answered 2022-07-06 Author has 13 answers
I suppose that the integrating factor I is defined by
I ( t ) = exp ( t 0 t a 1 ( s ) d s )
and hence has the property
I ( t ) = exp ( t 0 t a 1 ( s ) d s ) a 1 ( t ) = I ( t ) a 1 ( t )
so
d d t ( x ( t ) I ( t ) ) = d d t x ( t ) I ( t ) + x ( t ) d d t I ( t ) = d d t x ( t ) I ( t ) + x ( t ) a 1 ( t ) I ( t )
which is the left hand side.
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