anonymous
  • anonymous
Find y. y'-e^ysinx=0
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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bahrom7893
  • bahrom7893
another diff eqn?
anonymous
  • anonymous
yess
bahrom7893
  • bahrom7893
yay lol workin on it.

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bahrom7893
  • bahrom7893
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
bahrom7893
  • bahrom7893
dy/dx - (e^y)(Sinx) = 0 dy/dx = (e^y)(Sinx)
bahrom7893
  • bahrom7893
Divide both sides by e^y and multiply by dx: dy/e^y = Sinx dx
bahrom7893
  • bahrom7893
For first integral: rewrite dy/e^y as e^(-y)dy, then let u = -y; du = -dy
bahrom7893
  • bahrom7893
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]
bahrom7893
  • bahrom7893
- Integral of (e^(-y)(-dy)) = - Integral of (e^u du) = - e^u = - e^(-y)
bahrom7893
  • bahrom7893
Int (dy/e^y)= Int (Sinx dx) - e^(-y) = - Cos(x) + C
bahrom7893
  • bahrom7893
multiply everything by -1 ( - 1 times a constant is still a constant so I will let -C be A) e^(-y) = Cos(x) + A
bahrom7893
  • bahrom7893
Take Ln of both sides: Ln(e^(-y)) = Ln(Cos(x)+A) -y = Ln(Cos(x)+A) y = - Ln(Cos(x)+A)

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