Consider the differential equation y'=x/(y-2). a. For which if the following initial values are you guaranteed to have a unique solution by Picard's Uniqueness Theorem? Justify your answer. I. y(0)=2 II.y(2)=0 B. Continue with the initial values found in a and using the statement of Picard's Uniqueness Theorem, choose an appropriate rectangle Ra,b and identify an epsilon>0 such that you can guarantee a unique solution y=y(x) on x€ ,[x0 - epsilon, x0 - epsilon] C. With the values of x0, Ra,b and epsilon, can you say something about the value of y(x0 + epsilon)? Do not solve the IVP, just use Picards Uniqueness Theorem
Consider the differential equation y'=x/(y-2).
a. For which if the following initial values are you guaranteed to have a unique solution by Picard's Uniqueness Theorem? Justify your answer.
I. y(0)=2 II.y(2)=0
B. Continue with the initial values found in a and using the statement of Picard's Uniqueness Theorem, choose an appropriate rectangle Ra,b and identify an epsilon>0 such that you can guarantee a unique solution y=y(x) on x€ ,[x0 - epsilon, x0 - epsilon]
C. With the values of x0, Ra,b and epsilon, can you say something about the value of y(x0 + epsilon)? Do not solve the IVP, just use Picards Uniqueness Theorem
D. Set x1 =x0 +epsilon and now consider the IVP with initial value y(x1)= y(x0 + epsilon). Can you repeat the above process I.e. use Picard's Uniqueness Theorem to find epsilon1 >0 such that y=y(x) is unique for x€[x0 - epsilon, x0 - epsilon]?
E. What is happening here? Why is this interesting?
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