solve-the-eqn-dr-d-r-a-2-r-2-a-2-r-2-cot-hint-let-a-2-r-2-a-2-r-2-2r-2-

Question Number 14630 by Don sai last updated on 03/Jun/17

solve the eqn

dr / d θ = [r (a^{2} - r^{2}) / a^{2} + r^{2}] \cot θ

hint . let a^{2} + r^{2} = a^{2} - r^{2} + {2 r}^{2} .

Commented by prakash jain last updated on 03/Jun/17

\frac{d r}{d θ} = \frac{r (a^{2} - r^{2})}{(a^{2} + r^{2})} \cot θ

\frac{(a^{2} + r^{2})}{r (a^{2} - r^{2})} d r = \cot θ d θ

\int \frac{(a^{2} + r^{2})}{r (a^{2} - r^{2})} d r

(u s i n g h i n t a^{2} + r^{2} = a^{2} - r^{2} + 2 r^{2}

= \int \frac{(a^{2} - r^{2} + 2 r^{2})}{r (a^{2} - r^{2})} d r

= \int \frac{(a^{2} - r^{2})}{r (a^{2} - r^{2})} d r + \int \frac{2 r^{2}}{r (a^{2} - r^{2})} d r

= \int \frac{d r}{r} + \int \frac{2 r}{(a^{2} - r^{2})} d r

s u n s t i t u t e a^{2} - r^{2} = u i n s e c o n d

p a r t - 2 r d r = d u

= \ln r + \int \frac{- d u}{u} + c

= \ln r - \ln u = l n \frac{r}{a^{2} - r^{2}} + C

\int \cot θ d θ = \ln \sin θ + c

\ln \frac{r}{a^{2} - r^{2}} = \ln \sin θ + c

\frac{r}{(a^{2} - r^{2}) \sin θ} = e^{c}

Commented by Don sai last updated on 06/Jun/17

Thank you

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