problem statement : the general solution of a first order linear differential equation is a family of curves of the form y=cf(x)+g(x). Show, conversely, that the differential equation of any such family is linear.
My attempt:
I took the derivative y'=cf'(x)+g'(x). I then separater c from the original equation and substituted it here, and the answer is linear. But this proves that it "can" have a differential equation that is linear. How does this prove that the only possible differential equation for such a family is linear?
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