First slide

Methods of solving first order first degree differential equations

Question

Solve the differential equation $(1 - x y + x^{2} y^{2}) d x = x^{2} d y$

Moderate

A

$\frac{1}{x} \tan (\ln |C x|)$
B

$\frac{1}{x} \cot (\ln |C x|)$
C

$\frac{1}{x} \sin (\ln |C x|)$
D

$\frac{1}{x} \cos (\ln |C x|)$

Solution

Given differential equation is $(1 - x y + x^{2} y^{2}) d x = x^{2} d y$
$\Rightarrow (1 - (x y) + {(x y)}^{2}) d x = x^{2} d y . . . . . . . (i)$
$\begin{array}{l} Substituting xy = v \Rightarrow y = \frac{v}{x} and \frac{d y}{d x} = (\frac{x \frac{d v}{d x} - v}{x^{2}}) \\ i.e., x^{2} d y = x d v - v d x in (i), we have \\ \begin{aligned} (1 - v + v^{2}) d x = x d v - v d x \\ \Rightarrow & (1 + v^{2}) d x = x d v \Rightarrow \int \frac{d v}{1 + v^{2}} = \int \frac{d x}{x} \\ \Rightarrow & \tan^{- 1} v = \ln | x | + \ln C \\ \Rightarrow & \tan^{- 1} v = \ln | C x | \Rightarrow v = \tan \ln | C x | \\ \Rightarrow & x y = \tan \ln | C x | \Rightarrow y = \frac{1}{x} \tan (\ln | C x |) \end{aligned} \end{array}$

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