Mathematics 82 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

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