First slide

Methods of solving first order first degree differential equations

Question

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

Moderate

A

$\tan x y = \log |c x|$
B

$\tan (\frac{x}{y}) = \log |c x|$
C

$x y = \tan (\log |c x|)$
D

$x + y = x y \tan x y + c$

Solution

$\begin{array}{l} Given equation is (1 - x y + x^{2} y^{2}) d x = x^{2} d y \\ \Rightarrow (1 + x^{2} y^{2}) d x = x (y d x + x d y) \\ \Rightarrow \frac{y d x + x d y}{1 + x^{2} y^{2}} = \frac{d x}{x} \\ \Rightarrow \frac{d (x y)}{1 + (x y)^{2}} = \frac{d x}{x} \\ Integrating on both sides \\ \Rightarrow \int \frac{d (x y)}{1 + (x y)^{2}} = \int \frac{d x}{x} \\ \Rightarrow \tan^{- 1} (x y) = \log | x | + \log | C | \\ \Rightarrow \tan^{- 1} (x y) = \log | x C | \\ \Rightarrow (x y) = \tan (\log | x C |) \end{array}$

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