If differentiable functions y = F(x) and y = G(X) both solve the initial value p

Question

If differentiable functions y = F(x) and y = G(X) both solve the initial value problem dy/dx = f(x), y(x_0) = y0, on an interval I, must F(x) = G(x) for every x in l? Justify the answer. A) F(x) = G(x) for every x in l because integrating f(x) results in one unique function. F(x) and G(x) are not unique. There are infinitely many functions that solve the initial value problem. When B) Solving the problem there is an integration constant C that can be any value. F(x) and G(x) could each have a different constant term. C) F(x) = G(x) for every x in l because when given an initial condition, we can find the integration constant when integrating f(x). Therefore, the particular solution to the initial value problem is unique. D) There is not enough information given to determine if F(x) = G(x).

Explanation / Answer

We know that , from The Existence and Uniqueness Theorem (of the solution a first order linear equation initial value problem) If the functions p and g are continuous on the interval I: < t < containing the point t = t0, then there exists a unique function y = (t) that satisfies the differential equation y + p(t) y = g(t) for each t in I, and that also satisfies the initial condition y(t0) = y0, where y0 is an arbitrary prescribed initial value.

hence F(x) =G(x) for every x in I because integrating f(x) results in one unique function.

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