We know that every differential equation is equivalent to a first-order system. I am trying to prove

Izabella Ponce

Izabella Ponce

Answered question

2022-06-20

We know that every differential equation is equivalent to a first-order system. I am trying to prove or disprove the converse. For example in R 2 , if we have a system x ˙ = f ( x , y ), y ˙ = g ( x , y ). Can we always convert it to one differential equation (for example, only in terms of x)? Under what condition, this is possible?

Answer & Explanation

Abigail Palmer

Abigail Palmer

Beginner2022-06-21Added 30 answers

Consider your example. In order to make this a second-order equation in x, you want to solve x ˙ = f ( x , y ) for y as a function of x and x ˙ , say y = a ( x , x ˙ ). This may or may not be possible, (and in most cases even if it is possible in principle it can't be done in closed form). Then we get
x ¨ = f 1 ( x , y ) x ˙ + f 2 ( x , y ) y ˙ = f 1 ( x , a ( x , x ˙ ) ) x ˙ + f 2 ( x , a ( x , x ˙ ) ) g ( x , a ( x , x ˙ ) )
where f 1 and f 2 are the partial derivatives of f.

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