how to integrate sec x

[sozener1]

y' = sec y

I'm supposed to use separation of variables to find general solution of the above differential equation

do you even need to separate variables here

don't you just take integral of both sides and get that y=ln sex y + tan y + c??

but in the answers it says it is y= sin^(-1)(x+c)

 

[stapel]

y' = sec y

I'm supposed to use separation of variables to find general solution of the above differential equation

do you even need to separate variables here

don't you just take integral of both sides....

You're saying that you want to do this:

. . . . .\(\displaystyle \int\, \dfrac{dy}{dx}\, dy\, =\, \int\, \sec(y)\, dx\)

But, with respect to x, the right-hand-side's integrand is a constant! Is this what you mean to be doing?

 

[sozener1]

You're saying that you want to do this:

. . . . .\(\displaystyle \int\, \dfrac{dy}{dx}\, dy\, =\, \int\, \sec(y)\, dx\)

But, with respect to x, the right-hand-side's integrand is a constant! Is this what you mean to be doing?

yes that's what I think I'm supposed to do

can you explain it??

 

[Deleted member 4993]

y' = sec y

I'm supposed to use separation of variables to find general solution of the above differential equation

do you even need to separate variables here

don't you just take integral of both sides and get that y=ln sex y + tan y + c?? .... this is incorrect

but in the answers it says it is y= sin^(-1)(x+c)

y' = sec(y)

\(\displaystyle \displaystyle \int \dfrac{dy}{sec(y)} = \int dx\)

\(\displaystyle \displaystyle \int cos(y) dy = x + c \)

Now continue.....

 

[HallsofIvy]

yes that's what I think I'm supposed to do

can you explain it??

No, that is NOT what you are supposed to do! You cannot have "dx" on one side of equation and "dy" on the other.


You could write the equation as either \(\displaystyle \int \frac{dy}{dx} dx= \int sec(y)dx\) or \(\displaystyle \int \frac{dy}{dx} dy= \int sec(y)dy\), but with the first, you cannot integrate \(\displaystyle \int sec(y) dx\) and with the second you cannot integrate \(\displaystyle \int \frac{dy}{dx} dy\).