First slide

Types of solutions of differential equations

Question

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

Moderate

A

$\frac{x y}{2} + \sin (\frac{x}{y}) + c = 0$
B

$\frac{{(x y)}^{2}}{2} + \sin (\frac{x}{y}) + c = 0$
C

$\frac{{(x y)}^{2}}{2} + \cos (\frac{x}{y}) + c = 0$
D

$x y + \tan (\frac{x}{y}) + c = 0$

Solution

Given differential equation is
$\begin{array}{l} \sin (\frac{x}{y}) (y d x - x d y) = x y^{3} (x d y + y d x) \\ \Rightarrow \sin (\frac{x}{y}) (\frac{y d x - x d y}{y^{2}}) = x y (x d y + y d x) \\ \Rightarrow d (- \cos (\frac{x}{y})) = \frac{1}{2} d (x y)^{2} \\ \Rightarrow (- \cos (\frac{x}{y})) = \frac{1}{2} (x y)^{2} \\ \Rightarrow - \cos (\frac{x}{y}) = \frac{1}{2} (x y)^{2} + c \end{array}$
$\Rightarrow \frac{{(x y)}^{2}}{2} + \cos (\frac{x}{y}) + c = 0$

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