First slide

Methods of solving first order first degree differential equations

Question

$The solution of differential equation 3 x (1 - x^{2}) y^{2} \frac{d y}{d x} + (2 x^{2} - 1) y^{3} = a x^{3} is$

Moderate

A

$x^{3} = a y + c y \sqrt{1 - x^{2}}$
B

$y^{3} = a x + c x \sqrt{1 - x^{2}}$
C

$x^{3} = a x + c x \sqrt{1 - x^{2}}$
D

$y^{3} = a y + c x \sqrt{1 - x^{2}}$

Solution

$\begin{array}{l} Put y^{3} = v \Rightarrow 3 y^{2} \frac{d y}{d x} = \frac{d v}{d x} \\ \Rightarrow \frac{d v}{d x} + \frac{2 x^{2} - 1}{x (1 - x^{2})} v = \frac{a x^{2}}{1 - x^{2}} \\ I. F = \frac{1}{x \sqrt{1 - x^{2}}} \\ G . S . i s v I F = \int Q . I F d x \\ \frac{y^{3}}{x \sqrt{1 - x^{2}}} = \int \frac{a x^{2}}{1 - x^{2}} \frac{1}{x \sqrt{1 - x^{2}}} d x \\ G.S is y^{3} = α x + c x \sqrt{1 - x^{2}} \end{array}$

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