FP1 First Order Differential Equation

insparato

Find the general solution of the differential equation

Unparseable latex formula:

\large \tex \frac{dy}{dx} - 2ycosecx = tan \frac{x}{2}

I got the integration factor as

Unparseable latex formula:

\large \tex (cosecx + cotx}^2

which turns up something nasty when you multiply it.

Some help would be appreciated.

Reply 1

silent ninja

inegrating factor e^(int -2cosecx)= e^(-2ln(tan0.5x)0= (tan0.5x)^-2

you should be able to do it from there.

Edit: int (cosecx) is in the formula book and there are two possibilities.

Reply 2

meef cheese

You don't need to multiply it out.

multiply both sides of the equation by the integrating factor to get:

(cosecx + cotx)^2 \frac{dy}{dx} - 2y(cosecx + cotx)^2 = (cosecx + cotx)^2 tan(\frac{x}{2}) \\ \frac{d}{dx}[y(cosecx + cotx)^2] = (cosecx + cotx)^2 tan(\frac{x}{2})

then just think of a way to integrate the RHS w.r.t. x

EDIT: or just do it the above way...

Reply 3

insparato

Am i suppose to know the half angle formula?

Nevermind, Guess my book never told me the integral of cosec.

Reply 4

insparato

I got it in the end.

Hmm, am i missing formula or something?

Reply 5

meef cheese

insparato

I got it in the end.

Hmm, am i missing formula or something?

\huge \int cosec x \; dx = -ln| cosec x + cot x|\; +\; C = ln|tan \frac{x}{2}| \; +\; C

The second one is in the Edexcel formulae booklet, but not in the Heinemann books.

Reply 6

insparato

Thanks, have to know your integration with these.

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FP1 First Order Differential Equation

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