Question

Find y. y'-e^ysinx=0

Answer 1

another diff eqn?

Answer 2

yess

Answer 3

yay lol workin on it.

Answer 4

FIRST OF ALL, In ALL Differential equations, especially if they are in terms of x and y, not only one variable, rewrite y' as dy/dx

Answer 5

dy/dx - (e^y)(Sinx) = 0
dy/dx = (e^y)(Sinx)

Answer 6

Divide both sides by e^y and multiply by dx:
dy/e^y = Sinx dx

Answer 7

For first integral:
rewrite dy/e^y as e^(-y)dy, then let u = -y; du = -dy

Answer 8

So:
Integral of (e^(-y)dy) = - Integral of (e^(-y)(-dy)) [I multiplied by two minuses ( double negative is a positive) to get a -dy=du]

Answer 9

- Integral of (e^(-y)(-dy)) = - Integral of (e^u du) = - e^u = - e^(-y)

Answer 10

Int (dy/e^y)= Int (Sinx dx)
- e^(-y) = - Cos(x) + C

Answer 11

multiply everything by -1 ( - 1 times a constant is still a constant so I will let -C be A)
e^(-y) = Cos(x) + A

Answer 12

Take Ln of both sides:
Ln(e^(-y)) = Ln(Cos(x)+A)
-y = Ln(Cos(x)+A)
y = - Ln(Cos(x)+A)