Given the first integrals, how do you find a differential equation that satisfies them?

Answer 1

Not sure what you are asking, but if you start with an equation, say

y^2=x^2+5

differentiate it

2yy'=2x implies ydy/dx=x implies ydy=xdx

integrate

1/2y^2+c1=1/2x^2+c2 implies y^2=x^2+c3

we know from the first equation c3=5

therefore

y^2=x^2+5

Just a circle from implicit differentiation to an implicit solution to a first-order non-linear non-homogeneous separable variable ordinary differential equation