First slide

Methods of solving first order first degree differential equations

Question

$The curve satisfy the differential equation (1 - x^{2}) y^{1} + x y = a x are$

Moderate

A

hyperbola
B

ellipse and parabola
C

ellipse and straight line
D

circle and ellipse

Solution

$\frac{d y}{d x} + \frac{x}{1 - x^{2}} y = \frac{a x}{1 - x^{2}}$
$\begin{array}{l} I. F = \int_{e} \frac{x}{1 - x^{2}} = \frac{1}{\sqrt{1 - x^{2}}} \\ If - 1 < x < 1 \\ Solution is y (\frac{1}{\sqrt{1 - x^{2}}}) = \int \frac{a x}{{(1 - x^{2})}^{3 / 2}} = \frac{a}{\sqrt{1 - x^{2}}} + c \\ \Rightarrow y = a + c \sqrt{1 - x^{2}} if - 1 < x < 1 \\ If x^{2} > 1 \\ - (y - a)^{2} - c^{2} x^{2} = c^{2} deep hyp \end{array}$

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