First slide

Methods of solving first order first degree differential equations

Question

$The differential equation \frac{d x}{d y} = \frac{3 y}{2 x} represents a family of hyperbolas (except when it represents a pair of lines) with$ $eccentricity$

Difficult

A

$\sqrt{\frac{8}{5}}$
B

$\sqrt{\frac{5}{3}}$
C

$\sqrt{\frac{2}{5}}$
D

$\sqrt{\frac{5}{2}}$

Solution

$\frac{d x}{d y} = \frac{3 y}{2 x}$
$\begin{array}{lrlrl} \Rightarrow \int 2 x d x = \int 3 y d y \\ \Rightarrow x^{2} = \frac{3 y^{2}}{2} + c \end{array}$
$or \frac{x^{2}}{3} - \frac{y^{2}}{2} = \frac{c}{3}$
$If c is positive, then e = \sqrt{1 + \frac{2}{3}} = \sqrt{\frac{5}{3}}$
$If c is negative, then e = \sqrt{1 + \frac{3}{2}} = \sqrt{\frac{5}{2}}$

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