y(dy/dx) − (y/(dy/dx)) = 2a a is a real number


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Question Number 62869 by aliesam last updated on 26/Jun/19

$y \frac{d y}{d x} - \frac{y}{\frac{d y}{d x}} = 2 a$ $a i s a r e a l n u m b e r$
	Answered by Hope last updated on 26/Jun/19

	$y {(\frac{d y}{d x})}^{2} - y = 2 a (\frac{d y}{d x})$ ${y p}^{2} - 2 a p - y = 0$ $p = \frac{2 a \pm \sqrt{4 a^{2} + 4 y^{2}}}{2 y}$ $\frac{d y}{d x} = \frac{2 a \pm \sqrt{4 a^{2} + 4 y^{2}}}{2 y}$ $\frac{y d y}{a \pm \sqrt{a^{2} + y^{2}}} = d x$ $t^{2} = a^{2} + y^{2} \to t d t = y d y$ $n o w \int \frac{y d y}{a + \sqrt{a^{2} + y^{2}}} = \int d x \to [c o n s i d e r i n g + s i g n]$ $\int \frac{t d t}{a + t} = \int d x$ $\int \frac{a + t - a}{a + t} d t = \int d x$ $\int d t - a \int \frac{d t}{a + t} = \int d x$ $t - a l n (a + t) = x + c$ $\sqrt{a^{2} + y^{2}} - a l n (a + \sqrt{a^{2} + y^{2}}) = x + c$ $i f c o s i d e r - v e s i g n$ $\int \frac{y d y}{a - \sqrt{a^{2} + y^{2}}} = \int d x$ $\int \frac{t d t}{a - t} = \int d x$ $\int \frac{a - t - a}{a - t} d t = - \int d x$ $\int d t - a \int \frac{d t}{a - t} = - \int d x$ $\int d t + \int \frac{a d t}{t - a} = - \int d x$ $t + a l n (t - a) = - x + c_{1}$ $\sqrt{a^{2} + y^{2}} + a l n (\sqrt{a^{2} + y^{2}} - a) = - x + c_{1}$
	Commented by aliesam last updated on 26/Jun/19

	$t h a n k y o u s i r b r i l l i a n t s o l$
	Commented by Hope last updated on 26/Jun/19

	$\frac{1}{2 \sqrt{a^{2} + y^{2}}} \times 2 y \frac{d y}{d x} + \frac{a}{\sqrt{a^{2} + y^{2}} - a} \times \frac{1}{2 \sqrt{a^{2} + y^{2}}} \times 2 y \frac{d y}{d x} = - 1$ $\frac{y \frac{d y}{d x}}{\sqrt{a^{2} + y^{2}}} (1 + \frac{a}{\sqrt{a^{2} + y^{2}} - a}) = - 1$ $\frac{y \frac{d y}{d x}}{\sqrt{a^{2} + y^{2}}} (\frac{\sqrt{a^{2} + y^{2}}}{\sqrt{a^{2} + y^{2}} - a}) = - 1$ $y \frac{d y}{d x} = - (\sqrt{a^{2} + y^{2}} - a)$ $\frac{d y}{d x} = \frac{- (\sqrt{a^{2} + y^{2}} - a)}{y}$ $y \frac{d y}{d x} - \frac{y}{\frac{d y}{d x}}$ $\frac{- (\sqrt{a^{2} + y^{2}} - a)}{1} + \frac{y^{2}}{\sqrt{a^{2} + y^{2}} - a}$ $\frac{y^{2} - (a^{2} + y^{2} + a^{2} - 2 a \sqrt{a^{2} + y^{2}})}{\sqrt{a^{2} + y^{2}} - a}$ $= \frac{2 a (\sqrt{a^{2} + y^{2}} - a)}{\sqrt{a^{2} + y^{2}} - a}$ $= 2 a p r o v e d$ $s o w e c a n g e t . . .$
	Commented by mr W last updated on 26/Jun/19

	$v e r y n i c e s i r!$
	Commented by Hope last updated on 26/Jun/19

	$t h a n k y o u s i r . . .$
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