How is y'(x)=f(x,y(x)) is the most general first order ordinary

Arslan Lyons

Arslan Lyons

Answered question


How is y(x)=f(x,y(x)) is the most general first order ordinary differential equation ? Isn't f(x,y(x),y(x))=0 the most general first order ODE ? I mean isn't a restriction to single out y(x) from f? Thank you for your help!

Answer & Explanation

Riccardo Stewart

Riccardo Stewart

Beginner2022-02-16Added 3 answers

Yes, the implicit form is more general. However, either one can isolate the first derivatives to the explicit form or the equation is not an ODE but an DAE (differential-algebraic equations) of positive index.
So in treating ODE theory it is sufficient and, via the Picard integral equation, also more expedient to treat the explicit form.

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