1

Now suppose that yo € R and v' (c) = F(v(z),2) , v(0) = yo has unique solution U: [0, 1] = R and that Un (0) = yo: Prove that un 57 uniformly. (Hint: Sin...

Question

Now suppose that yo € R and v' (c) = F(v(z),2) , v(0) = yo has unique solution U: [0, 1] = R and that Un (0) = yo: Prove that un 57 uniformly. (Hint: Since uniform convergence is given metric, previous homework problem shows that it is enough to show that given any subsequence Unk there is a further subsequence Unki with Unki 4 uniformly: To prove this, argue that the Arzela-Ascoli theorem applies to {un n € N} and use uniqueness of the solution of the ODE): (d) Now suppose that for a

Now suppose that yo € R and v' (c) = F(v(z),2) , v(0) = yo has unique solution U: [0, 1] = R and that Un (0) = yo: Prove that un 57 uniformly. (Hint: Since uniform convergence is given metric, previous homework problem shows that it is enough to show that given any subsequence Unk there is a further subsequence Unki with Unki 4 uniformly: To prove this, argue that the Arzela-Ascoli theorem applies to {un n € N} and use uniqueness of the solution of the ODE): (d) Now suppose that for all y € R, the differential equation v (c) = F(v(x),x) , v(0) = y has a unique solution %y: [0,1] = R Explain why part (c) implies that the map R + C([0, 1],R) given by y + %y is continuous when we give C([0,1], R) the metric induced by the supremum norm_



Answers

Existence and Uniqueness. Under the assumptions of Theorem 1, we will prove that equation (8) gives a solution to equation (4) on $ (a, b) $. We can then choose the constant C in equation (8) so that the initial value problem (15) is solved.

(a) Show that since $ P(x) $ is continuous on $ (a, b) $, then $ \mu(x) $ defined in (7) is a positive, continuous function satisfying $ d\mu/dx = P(x)\mu(x) $ on $ (a, b) $.

(b) Since $ \frac{d}{d x} \int \mu(x) Q(x) d x=\mu(x) Q(x) $, verify that y given in equation (8) satisfies equation (4) by differentiating both sides of equation (8).

(c) Show that when we let $ \int \mu(x) Q(x) d x $ be the antiderivative whose value at x0 is 0 (i.e.,
$ \int_{x_{0}}^{x} \mu(t) Q(t) d t ) $) and choose C to be $ y_{0} \mu\left(x_{0}\right) $, the initial condition $ y\left(x_{0}\right)=y_{0} $ is satisfied.

(d) Start with the assumption that $ y(x) $ is a solution to the initial value problem (15) and argue that
the discussion leading to equation (8) implies that $ y(x) $ must obey equation (8). Then argue that the initial condition in (15) determines the constant C uniquely.


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