(a) for what values of r does the function y = erx satisfy the differential equation 5y'' + 34y' − 7y = 0? (enter your answers as a comma-separated list.) r = ) if r1 and r2 are the values of r that you found in part (a), show that every member of the family of functions y = aer1x + ber2x is also a solution. (let r1 be the larger value and r2 be the smaller value.)

a) r=-7 and r=1/5

b) See proof below

Step-by-step explanation:

Let y(x) be the function

Then

a)

Replacing these values in the differential equation

5y'' + 34y' − 7y = 0

we get

dividing both sides by we get the quadratic equation

whose solutions are r = -7 and r=1/5

b)

We must show that for any constants a, b, the function

is also a solution of the differential equation

We have

So

Hence

is also a solution for any values of a and b.

(d)

step-by-step explanation:

given: the line in the graph passes through points (5,30) and (6,26).

then, the equation of line is:

now, put y=a(n) and x=n,

hence, option d is correct.