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

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional