Question

solve the differential equation dy/dx=-csc^2(5x)/(cot(5x)) for the particular solution that passes through the point (pi/20,3)

Answer 1

\[\frac{dy}{dx}=-\frac{\csc^2(5x)}{\cot(5x)},y(\frac\pi{20})=3\]

Answer 2

Simplify the stuff by expressing everything in terms of sine and cosine first

Answer 3

what would the first step be? im confused on how to start it

Answer 4

Simplify \(-\dfrac{\csc^2(5x)}{\cot(5x)}\) first

Answer 5

i dont know how to simplify that :/

Answer 6

By expressing everything in terms of sine and cosine

Answer 7

\[\tan\theta=\frac{\sin\theta}{\cos\theta}\]\[\cot\theta=\frac{\cos\theta}{\sin\theta}\]\[\csc\theta=\frac1{\sin\theta}\]\[\sec\theta=\frac1{\cos\theta}\]

Answer 8

\[\frac{\csc^2(\theta)}{\cot(\theta)}=\frac{1/\sin^2(\theta)}{\cos(\theta)/\sin(\theta)}=\]

simplify this by multiplying the numerator and denominator by sine theta

Answer 9

For example, \(\cot(5x)=\dfrac{\cos(5x)}{\sin(5x)}\)

Answer 10

whats next?

Answer 11

Follow his step

Answer 12

\[\frac{ 1 }{ \sin(5x) }\]/cos(5x)

Answer 13

Use this Product to Sum formula
in the numerator, to get it into a single trig function\[\sin\psi\cos\phi=\tfrac12[\sin(\psi+\phi)+\sin(\psi-\phi)]\]

Answer 14

*denominator

Answer 15

.... @iTzVinny21