find to the nearest integer the value of C that satisfies (y-6)dy=(x+4)/(y-3)dx, y(-1/12)=2

(y-6)dy=(x+4)/(y-3)dx

(y^2 - 9y + 18) dy = (x+4) dx
1/3 y^3 - 9/2 y^2 + 18y = 1/2 x^2 + 4x + C

plug in (-1/12,2)

8/3 - 18 + 36 = 1/2 (1/144) + 4 (-1/12) + C
C = 6047/288

That's my final answer, but you'd better check it.

yes cool um didn't really help me but cool way of answering.

hrjetgfauehgth

what is the integer values of n that satisfies this inequality -5<3n<12?

Please answer :)

21

To find the value of C that satisfies the given differential equation, we can use the method of separation of variables.

1. Start by rearranging the equation to isolate the variables: (y - 6) dy = (x + 4)/(y - 3) dx
2. Multiply both sides of the equation by (y - 3) to get rid of the denominators: (y - 6)(y - 3) dy = (x + 4) dx
3. Expand the left side of the equation: (y^2 - 9y + 18) dy = (x + 4) dx
4. Integrate both sides of the equation with respect to their respective variables:
∫(y^2 - 9y + 18) dy = ∫(x + 4) dx
∫y^2 dy - 9∫y dy + 18∫dy = ∫(x + 4) dx
(y^3/3 - 9y^2/2 + 18y) = (x^2/2 + 4x) + C

5. Now, we can use the given initial condition y(-1/12) = 2 to find the value of C.
Substitute y = 2 and x = -1/12 in the equation:
(2^3/3 - 9(2)^2/2 + 18(2)) = ((-1/12)^2/2 + 4(-1/12)) + C
Simplify the equation: 2/3 - 18/2 + 36 = 1/288 + 4/12 + C
Combine the terms: 2/3 - 9 + 36 = 1/288 + 12/288 + C
Simplify the fractions: 2/3 + 27/288 + C = 148/288
Combine the fractions: 2/3 + 1/12 + C = 37/72
Convert to a common denominator: 16/24 + 2/24 + C = 37/72
Combine the terms: 18/24 + C = 37/72
Convert the fractions to a common denominator: 36/48 + C = 37/72
Combine the fractions: 36/48 + C = 37/72
Now, subtract 36/48 from both sides: C = 37/72 - 36/48
Convert the fractions to a common denominator and calculate: C = (37*48 - 36*72)/(72*48)
Simplify: C = (1776 - 2592)/(3456)
Calculate: C = -816/3456
Simplify the fraction: C = -17/72

Therefore, the value of C that satisfies the given differential equation is -17/72. Rounding it to the nearest integer, we get C ≈ -1.