# Integrating factor confusion

for the question, siny+cosydy=0, i want to find an integrating factor.
my work:
(1/F)(dF/dx)=(1/cosy)(cosy+siny)=1+tany
=>lny=x +xtany +c`
=> y =ce^(x+xtany)
however, the question wants the integrating factor to be e^x...
why?

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HallsofIvy
Homework Helper
Can I assume you mean sinydx+ cosydy= 0? Without a dx in there, it doesn't make sense. If that's the case, then an obvious integrating factor is 1/siny since multiplying through by that gives dx+ (cosx/sinx)dy= 0 which is clearly exact.

I don't know what you mean by "the question wants the integrating factor to be e^x"!
I wasn't aware that questions wanted anything!

Isn't that equation seperable?

opps!!! i'm sorry for the mistype! :P
you're right, it's "sinydx+ cosydy= 0"
that question wanted to prove that the integrating factor is e^x, but the integrating factor that i found was y =ce^(x+xtany)...

lurflurf
Homework Helper
so you want an integrating factor u such that
$$\frac{\partial}{\partial y}u\sin(y)=\frac{\partial}{\partial x}u\cos(y)$$
or
$$\frac{\partial u}{\partial y}\sin(y)+u\cos(y)=\frac{\partial u}{\partial x}\cos(y)$$
integrating factors are not unique so assume
$$\frac{\partial u}{\partial y}=0$$

HallsofIvy
Homework Helper
If the problem says "show that ex is an integrating factor", thenyou don't have to find the integrating factor yourself (as lurflurf said, integrating factors are not unique), just multiply the equation by ex and see if the result is exact.

If you got ce^(x+xtany) as an integrating factor, you sure like doing things the hard way! As I said earlier, 1/sin y is an obvious integrating factor (because, as Corneo said, the equation is separable. Multiplying by
1/sin y "separates" it)

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lol...
i didn't think of that...
thank you very much!!! :)