Question

Use of an integrating factor:
"Find the general solution of the given differential equation. State an interval on which the general solution is defined.

\[cos(x) \frac{dy}{dx} + ysin(x) = 1\]

Answer 1

Is this one separable?

Answer 2

I started by dividing out the cos(x) to get \[\frac{dy}{dx} + ytan(x) = \frac{1}{cos(x)}\]
Then created an integrating factor: \[IF = e^{\int\limits_{}^{}P(x) dx} \rightarrow e^{\int\limits_{}^{}\tan(x) dx}\]

Answer 3

I do not believe it is, though I did not try and separate it.

Answer 4

After integrating I had \[e^{-\ln|\cos(x)|}\]
which I multiplied both sides of the equation by.
\[e^{-\ln|\cos(x)|} \frac{dy}{dx} + e^{-\ln|\cos(x)|} ytan(x) = e^{-\ln|\cos(x)|} \frac{1}{\cos(x)}\]

Answer 5

Not exactly sure where to go from here.

Answer 6

Refer to the attachment, a screen capture .

Answer 7

@eSpeX you still there?

Answer 8

\[e^{-\ln\cos x}=\frac1{\cos x}=\sec x\]

Answer 9

@TuringTest Thank you, I was pulling my hair out because of that silly sec.