(b) Use the Picard-Lindelof Theorem to show that 0 < A <1 is required for the ODE in (a) with the initial condition y(0) = 1 to have a unique solution in a rectangular space D = {|x| ≤ A, |y − 1| ≤ B}. Sketch the position of this rectangle D in the xy plane. Find out all other conditions between A and B to guarantee the uniqueness of the solution in D.
Solve b and c with a step by step written answer please.
(a) Find the general solution of the following ordinary differential equation (x − 1)y' = 2y.
(b) Use the Picard-Lindelof Theorem to show that 0 < A <1 is required for the ODE in (a) with the initial condition y(0) = 1 to have a unique solution in a rectangular space D = {|x| ≤ A, |y − 1| ≤ B}. Sketch the position of this rectangle D in the xy plane. Find out all other conditions between A and B to guarantee the uniqueness of the solution in D.
(c) Use the Picard-Lindelof Theorem to show whether there exists a unique solution to the ODE in (a) with a different initial condition y(1) = 0. If not, based on the general solution obtained in (a) and this initial condition y(1) = 0, sketch and describe all possible solutions to this initial value problem in the xy plane.
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