First slide

Types of solutions of differential equations

Question

$The solution of \frac{x + y d y / d x}{y - x d y / d x} = \frac{x \sin^{2} (x^{2} + y^{2})}{y^{3}} is given by$

Moderate

A

$- \cot (x^{2} + y^{2}) = {(\frac{x}{y})}^{2} + c$
B

$\tan (x^{2} + y^{2}) = x^{2} y^{2} + c$
C

$\cot (x^{2} + y^{2}) = \frac{x}{y} + c$
D

$None of these$

Solution

$\frac{x + y \frac{d y}{d x}}{y - x \frac{d y}{d x}} = \frac{x \sin^{2} (x^{2} + y^{2})}{y^{3}}$
$\begin{array}{l} \frac{x d x + y d y}{y d x - x d y} = \frac{x \sin^{2} (x^{2} + y^{2})}{y^{3}} \\ \frac{x d x + y d y}{\sin^{2} (x^{2} + y^{2})} = \frac{x (y d x - x d y)}{y^{3}} \\ \Rightarrow d (- \frac{\cot (x^{2} + y^{2})}{2}) = \frac{x y d x - x^{2} d y}{y^{3}} = \frac{x}{y} [\frac{y d x - x d y}{y^{2}}] \\ \Rightarrow d (- \cot (x^{2} + y^{2})) = 2 \frac{x}{y} (\frac{d y x - x d y}{y^{2}}) \\ \Rightarrow d (- \cot (x^{2} + y^{2})) = d {(\frac{x}{y})}^{2} \\ \Rightarrow - \cot (x^{2} + y^{2}) = {(\frac{x}{y})}^{2} + c \end{array}$

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