First slide

Methods of solving first order first degree differential equations

Question

$The solution of the DE \cos x d y = y (\sin x - y) d x, 0 < x < \frac{π}{2} is$

Moderate

A

$tanx=(secx+c)y$
B

$secx=(tanx+c)y$
C

$ysecx=tanx+c$
D

$ytanx=secx+c$

Solution

$\begin{array}{l} \frac{dy}{dx} = ytanx - y^{2} secx \\ \Rightarrow \frac{1}{y^{2}} \frac{dy}{dx} - \frac{1}{y} \tan x = - \sec x \\ t = \frac{1}{y} \\ \frac{dt}{dx} = \cdot \frac{- 1}{y^{2}} \frac{dy}{dx} \\ \to \frac{- dt}{dx} - \tan xt = - \sec x \\ \to \frac{dt}{dx} + (\tan x) t = secx \end{array}$
$\begin{array}{l} I.F. = e^{\int \tan x d x} = \sec x \\ t s e c x = \int s e c^{2} x d x \\ tsecx = \tan x + c \\ \frac{1}{y} \sec x = \tan x + c \end{array}$

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