Question

4. Consider the differential equation with initial condition r(0) = 0 (a) What does the existence and uniqueness theorem tell you about the solution to this IVP? (10 points) (b) Use separation of variables to find the solution for the IVP r(to) = Io for to +0. (5 points) (c) Are the solutions to b) unique? (5 points) (d) Sketch solutions for Xo = --1,0,1 and to = 1 and show that for all to and to the solution goes through the point (x, t) = (0,0). Generalizing this sketch to arbitrary Lo, prove that the original IVP (2(0) = 0 ) has infinitely many solutions? (5 points)

Answer 1

l=fit, n = 7 Qoy Protchi kirine, f (t, 2) im not continmon in any nbad of to, no )= 10,0), the existence and uniquenen theorem has no conclusion about it.

(6) t.dx = x dt - " On integrating: [c= I.C] mca)= m(t) + me il, a=ct sein, it tato, na to, we get = 은 &lt) = ( 20 ) (6) sinine, as = ť and f and the bath are continuam in a wild R of a (do, to) [to to and both are bounded, the solution in unique [from existence and uniquenen theorem), (d) clearly the volutions of this family are all linen framing through dingin endeling the y-anir.

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