Mathematics 24 Online
OpenStudy (amistre64):

ODE: y' + y tan(x) = cos(x) another devious ploy im sure ;)

OpenStudy (anonymous):

Hello!

OpenStudy (amistre64):

howdy

OpenStudy (anonymous):

OpenStudy (amistre64):

ill brace myself

OpenStudy (amistre64):

no giving hints james lol

OpenStudy (amistre64):

unless they are cryptic of course

OpenStudy (jamesj):

Have you learnt integrating factors yet? If so, find it! That's maybe cryptic.

OpenStudy (anonymous):

y'(x)+y(x) tan(x) = cos(x) or y'[x] + y[x] Tan[x] == Cos[x]

OpenStudy (anonymous):

first order linear - use integrating factor?

OpenStudy (amistre64):

i know the e^int(...) stuff yes

OpenStudy (jamesj):

If you do, this ODE is fun and quite easy.

OpenStudy (anonymous):

IF = e^INT(tanx dx)

OpenStudy (amistre64):

good, what can we do with that?

OpenStudy (jamesj):

well, simplify

OpenStudy (jamesj):

...by actually evaluating the expression.

OpenStudy (amistre64):

essentially i believe we are turning it into a reversal of the product rule

OpenStudy (jamesj):

yes, that's the idea of the integrating factor. So stop stalling and just do it. :-)

OpenStudy (amistre64):

ok, if we can turn this into a x'y + xy' by utilizing the e^x stuff; multiply this contraption by: $$e^{\int tan(x)dx}$$ $y'e^{ln|sec(x)|}+tan(x)\ e^{ln|sec(x)|}=cos(x)\ e^{ln|sec(x)|}$ $ye^{ln|sec(x)|}=\int cos(x)\ e^{ln|sec(x)|}dx$

OpenStudy (amistre64):

since e^(ln|sec(x)|) = sec(x) ...

OpenStudy (amistre64):

$ye^{ln|sec(x)|}=\int cos(x)\ sec(x)dx$ $ye^{ln|sec(x)|}=\int \frac{cos(x)}{cos(x)}dx$ $ye^{ln|sec(x)|}=\int dx$ $ye^{ln|sec(x)|}=x$ right? so far

OpenStudy (amistre64):

forgot a +C as always tho

OpenStudy (amistre64):

$ye^{ln|sec(x)|}=x+C$ $y=\frac{x+C}{e^{ln|sec(x)|}}$ $y=\frac{x+C}{sec(x)}$ $y=x\cos(x)+C\cos(x)$ maybe

OpenStudy (jamesj):

Yep, that's it.

OpenStudy (amistre64):

yay!! i knew i could do it ... again lol

OpenStudy (amistre64):

taking diffy qs next term

OpenStudy (jamesj):

I'd just say that you can simplify the integrating factor to sec x much sooner rather than later.

OpenStudy (amistre64):

you can, but wheres the excitement in that

OpenStudy (jamesj):

Elegance is exciting.

OpenStudy (amistre64):

i dropped a "y" at the start but i picked it up in the end