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

i believe that the answer is y+6=1/3(x+2)