I am looking to solve the following equations numerically: a x = d

cyfwelestoi 2022-05-31 Answered
I am looking to solve the following equations numerically:
a x = d d t ( f ( x , y , t ) d y d t ) , b y = d d t ( g ( x , y , t ) d x d t )
For arbitrary functions f and g and constants a and b. I am struggling to find a way to transform this into a system of first order differential equations that I can pass into a solver. It looks like I will need to define these implicitly, but I'm not sure how to do that.
My best attempt so far is the following:
z 1 = f ( x , y , t ) d y d t z 2 = g ( x , y , t ) d x d t z 3 = a x z 4 = b y
( z 1 z 2 z 3 z 4 ) = ( 0 0 1 0 0 0 0 1 0 a g ( t , x , y ) 0 0 b f ( t , x , y ) 0 0 0 ) ( z 1 z 2 z 3 z 4 )
However, this seems fairly inelegant and assumes that you are always able to divide by f and g. I'm trying to keep this as general as possible, so don't want to make that assumption. Is there a better way to turn this into a system of first order differential equations implicitly?
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Answers (1)

redclick53
Answered 2022-06-01 Author has 12 answers
I would rather define u=x' and v=y'. Then, your system becomes
( x y f v g u ) = ( u v a x f x u v f y v 2 b y g x u 2 g y u v )
As long as f 0 g, this is a nonlinear system of first order ODEs. If, at a certain point, f=0, or g=0, or both, then this system is a differential-algebraic equation
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