First slide

Methods of solving first order first degree differential equations

Question

$The solution of differential equation \frac{d y}{d x} = \frac{\sqrt{1 - y^{2}}}{y} determines family of circles with$

Moderate

A

variable radii and fixed centre at $(1, 0)$
B

variable radii and fixed centre at $(0, - 1)$
C

fixed radius 1 and variable centre along $y -$ axis
D

fixed radius 1 and variable centre along $x -$ axis

Solution

$\begin{array}{l} We have, \frac{d y}{d x} = \frac{\sqrt{1 - y^{2}}}{y} ∣ \\ On integrating both sides \\ we get \int \frac{y}{\sqrt{1 - y^{2}}} d y = \int d x \\ \Rightarrow - \frac{1}{2} \int \frac{- 2 y}{\sqrt{1 - y^{2}}} d y = \int d x \\ \Rightarrow - \sqrt{1 - y^{2}} = x + c \end{array}$
$On squaring both sides, we get (x + c)^{2} + y^{2} = 1$
Which is the equation of circle whose fixed radius is 1 and centre is variable on $x -$ axis

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