(a) By inspection find a one-parameter family of solutions of the differential equation xy' = y. Verify that each member of the family is a solution of the initial-value problem xy' = y, y(0) = 0.

(b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy' = y would have a unique solution through a point (x₀, y₀) in R.

(c) Verify that the piecewise-defined function satisfies the condition y(0) = 0. 

 

Determine whether this function is also a solution of the initial value problem in part (a).

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