(a) Construct a linear first order differential equation of the form xy ' 3y g(x) for which y x 3 1 c x 3 is its general solution Give an interval I of definition of this solution (b) Give an initial condition y(x 0 ) y 0 for the DE found in part (a) so that the solution of the IVP is y x 3 1 x 3 Repeat if the solution is y x 3 2 x 3 Give an interval I of definition of each of these solutions Graph the solution curves Is there an initial value problem whose solution is defined on ( , ) (c) Is each IVP found in part (b) unique That is, can there be more than one IVP for which, say, y x 1 x 3 , x in some interval I, is the solution

Question: (a) Construct a linear first-order differential equation of the form xy' + 3y = g(x) for which y = x 3 1 c/x 3 is

(a) Construct a linear first-order differential equation of the form xy' + 3y = g(x) for which y = x³ 1 c/x³ is its general solution. Give an interval I of definition of this solution.

(b) Give an initial condition y(x₀) = y₀ for the DE found in part (a) so that the solution of the IVP is y = x³ – 1/x³. Repeat if the solution is y = x³ + 2/x³. Give an interval I of definition of each of these solutions. Graph the solution curves. Is there an initial-value problem whose solution is defined on (-∞, ∞)?

(c) Is each IVP found in part (b) unique? That is, can there be more than one IVP for which, say, y = x‑ - 1/x³, x in some interval I, is the solution?

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