First slide

Methods of solving first order first degree differential equations

Question

Solution of the differential equation $x \sin \frac{y}{x} d y = (y \sin \frac{y}{x} - x) d x$ is

Moderate

A

$\log x = \cos \frac{y}{x} + c$
B

$\log y = \cos \frac{y}{x} + c$
C

$\log x = \cos \frac{x}{y} + c$
D

none

Solution

$\begin{array}{l} We have \frac{d y}{d x} = \frac{y \sin \frac{y}{x} - x}{x \sin \frac{y}{x}} put y = V x \\ So that \frac{d y}{d x} = V + x \frac{d V}{d x} \\ Hence, V + x \frac{d V}{d x} = \frac{V \sin V - 1}{\sin V} = V - \frac{1}{\sin V} \\ \Rightarrow x \frac{d V}{d x} = - \frac{1}{\sin V} \Rightarrow \int \frac{d x}{x} + \int \sin V d V = c \\ \Rightarrow \log x - \cos V = c \Rightarrow \log x = \cos \frac{y}{x} + c \end{array}$

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