How to use differential equations to write x(t) in terms of y and y_0? {x'=-ax+bxy, y'=cy-dxy

Liberty Page 2022-09-25 Answered
How to use differential equations to write x ( t ) in terms of y and y 0 ?
{ x = a x + b x y y = c y d x y
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Answers (1)

Trace Arias
Answered 2022-09-26 Author has 6 answers
(1) d x ( t ) d t = x = x ( t ) ( a + y ( t ) b )
(2) d y ( t ) d t = y = y ( t ) ( c x ( t ) d )
(1)/(2) leads to:
(3) d x d y = x y a + y b c x d
or
(4) c x d x d x = a + y b y d y
Integration of (4) leads to
(5) c log x x d = a log y + b y + y 0 := A ( y , y 0 )
The solution to (5) can be expressed in terms of Lambert W function
(6) x = ( c / d ) W ( B ( y , y 0 ) )
where
B ( y , y 0 ) = ( d / c ) exp ( ( 1 / c ) A ( y , y 0 ) )
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