Question

4. (10pts) The differential equation $(2x - y) + (2y - x)y' = 0$ is exact. Solve the initial value problem $(2x - y) + (2y - x)y' = 0$ and $y(1) = 3$ 5. Consider the (autonomous) differential equation $y' = f(y) = y^2(1 - y)$. Do not solve the differential equation! (a) (5pts)Plot the graph of $f(y)$ and the phase line. (b) (5pts) Determine the equilibrium solutions and classify each of them by unstable, asymptotically stable, semistable. (c) (5pts) Describe the long term behavior (that is, the behavior as $t \to \infty$) of the solutions to the IVP for various values of $y(0) = y_0$. Give as complete a description as possible.

          4. (10pts) The differential equation $(2x - y) + (2y - x)y' = 0$ is exact. Solve the initial value problem $(2x - y) + (2y - x)y' = 0$ and $y(1) = 3$

5. Consider the (autonomous) differential equation $y' = f(y) = y^2(1 - y)$. Do not solve the differential equation!
(a) (5pts)Plot the graph of $f(y)$ and the phase line.
(b) (5pts) Determine the equilibrium solutions and classify each of them by unstable, asymptotically stable, semistable.
(c) (5pts) Describe the long term behavior (that is, the behavior as $t \to \infty$) of the solutions to the IVP for various values of $y(0) = y_0$. Give as complete a description as possible.

4. (10pts) The differential equation (2x - y) + (2y - x)y' = 0 is exact. Solve the initial value problem (2x - y) + (2y - x)y' = 0 and y(1) = 3

5. Consider the (autonomous) differential equation y' = f(y) = y^2(1 - y). Do not solve the differential equation!
(a) (5pts)Plot the graph of f(y) and the phase line.
(b) (5pts) Determine the equilibrium solutions and classify each of them by unstable, asymptotically stable, semistable.
(c) (5pts) Describe the long term behavior (that is, the behavior as t →∞) of the solutions to the IVP for various values of y(0) = y0. Give as complete a description as possible.

Added by Pamela F.

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