For the differential equations dydx=sqrt(y^2−36) does the existence/uniqueness theorem guarantee that there is a solution to this equation through the point

1. (−4,6)

2. (2,−6)

3. (−5,39)

4. (−1,45)

Please explain the process of obtaining the answer.

1. (-4,6) there is no a solution to the equation through this point

2. (2,−6) there is no a solution to the equation through this point

3. (−5,39) there is a solution to the equation through this point

4. (−1,45)  there is a solution to the equation through this point

Step-by-step explanation:

Using the existence and uniqueness theorem:

Let:

Now, let's find the domain of , due to the square root:

So the domain of the function is:

Now, due to the fraction the denominator must be also different from 0, so:

So, the theorem  tells us that for each there exists a  unique solution defined in an open interval around .

1. (-4,6)  there is no a solution to the equation through this point because

2. (2,−6)  there is no a solution to the equation through this point because

3. (−5,39) there is a solution to the equation through this point because

4. (−1,45)  there is a solution to the equation through this point because

b

step-by-step explanation:

the polynomial is a perfect square trinomial of the form a^2-2ab+b^2:

option a. true

step-by-step explanation:

9x^2-12x+4

comparing with the form a^2-2ab+b^2

1.) a^2=9x^2

solving for a: square root both sides of the equation:

sqrt(a^2)=sqrt(9x^2)

a=sqrt(9) sqrt(x^2)

a=3x

2.) b^2=4

solving for b: square root both sides of the equation:

sqrt(b^2)=sqrt(4)

b=2

to be a perfet square trinomial 2ab must be equal to 12x. let's check:

2ab = 2 (3x) (2)

2ab = 12x

correct