Solving Linear Equations: y' - ytanx = 2x/cosx

[Math Is Hard]

Hi, I am having a little trouble solving a linear equation:

$$y' cosx = ysinx + 2x$$

I translated it into standard y' + P(x)y= Q(x) format as

$$y' - ytanx = 2x/cosx$$

then I needed an integrating factor, so I used

$$I(x)=e^{-\int tanx\:dx} = cosx$$

when I multiply this to both sides of $$y' - ytanx = 2x/cosx$$
it doesn't seem to do much good. I just get back to where I started, and I am not seeing anything that will wrap into a nice neat little derivative on the LHS.

Can you show me where I am going wrong? Thanks!

 

[quantumdude]

Go back up to your first line. Instead of dividing by cos(x), bring the ysin(x) term to the LHS. Then you have:

y'cos(x)-ysin(x)=2x.

The LHS is identical to d(ycos(x))/dx.

 

[Math Is Hard]

oh, wow! I knew it wasn't supposed to be that hard!

so my solution is

$$y = x^2 / cos(x) + C$$

I think that's right..

Thanks, Tom!

 

[quantumdude]

$$y = x^2 / cos(x) + C$$

The "+C" should be over the cos(x), with the x².

Thanks, Tom!

No problemo.

 

[Math Is Hard]

Right you are! The devil's in the details!

point noted

 

[quantumdude]

By the way, what course is this for? It looks like Differential Equations, but I could have sworn that not too long ago you were asking questions about subject matter from Calculus II.

 

[Math Is Hard]

I am still wrapping up the second part of single variable calculus - I am done in two weeks - hooray!
UCLA is on a quarter system, so for engineering series there are two quarters of single variable, two quarters of multi-variable, and then a quarter of linear algebra and a quarter of ordinary diff. equations. (I think that's how it goes, anyway)
Toward the end of this class they give us a little taste of differential equations, Taylors, and some other things we might come across if we continue on in math. What's kinda weird though is that polar coordinates aren't covered until multivariable calculus at UCLA, and I had heard that it was standard to cover those in single variable calc.

 

[quantumdude]

Toward the end of this class they give us a little taste of differential equations, Taylors, and some other things we might come across if we continue on in math.

A-ha. Where I work, we teach a little taste of Diff Eq in Calculus I (but not even as heavy as what you have presented here), and Taylor series are done in great detail in Calculus II.

What's kinda weird though is that polar coordinates aren't covered until multivariable calculus at UCLA, and I had heard that it was standard to cover those in single variable calc.

Yes, we cover polar coordinates extensively in Calculus II (single variable), and we do it again in Calculus III (multivariable). But you can do polar coordinates from scratch in a multivariable setting.