Solve these linear equations in the form y=yn+yp with yn=y(0)e^at.
a. y'-4y=-8
b. y'+4y=8
which one has a steady state?

a) . It does not have a steady state

b) . It has a steady state.

Step-by-step explanation:

a)

The first step is finding . So:

We have to find the eigenvalues of this differential equation, which are the roots of this equation:

So:

Since this differential equation has a positive eigenvalue, it does not have a steady state.

Now as for the particular solution.

Since the differential equation is equaled to a constant, the particular solution is going to have the following format:

So

C is a constant, so (C)' = 0.

The solution in the form is

b)

The first step is finding . So:

We have to find the eigenvalues of this differential equation, which are the roots of this equation:

So:

Since this differential equation does not have a positive eigenvalue, it has a steady state.

Now as for the particular solution.

Since the differential equation is equaled to a constant, the particular solution is going to have the following format:

So

C is a constant, so (C)' = 0.

The solution in the form is

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step-by-step explanation: