Consider the autonomus equation $\frac{dy}{dt}=-2(y-1)(y-2)(y-a)^2$

Consider the autonomus equation $ \frac{dy}{dt}=-2(y-1)(y-2)(y-a)^2$ , where $ a$ is any real number.

Then,

$ (1)$ Plot the phase diagram showing the solution curves.

$ (2)$ show that new solution can be generated from the old solutions (in $ (a)$ ) by time shifting i.e, replacing $ y(t)$ by $ y(t-t_0)$ .

Answer:

$ (a)$ I have drawn the phase plot showing the solutions.

Please help me with the part $ (b)$ .

If we replace $ y(t)$ by $ y(t-t_0)$ , then we have

$ \frac{dy(t-t_0)}{dt}=-2(y(t-t_0)-1)(y(t-t_0)-2)(y(t-t_0)-a)^2$ .

How to confirm that we get a new solution?

Help me

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