First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $(y d x + x d y) x \cos (\frac{y}{x}) = (x d y - y d x) y \sin (\frac{y}{x})$ is

Moderate

A

$\frac{x}{y} \sec \frac{y}{x} = c x^{2}$
B

$\frac{y}{x} \cos \frac{y}{x} = c y^{2}$
C

$\frac{y}{x} \sin \frac{y}{x} = c x^{2}$
D

none

Solution

The equation can be written as
$\begin{array}{l} \Rightarrow (y + x \frac{d y}{d x}) x \cos \frac{y}{x} = (x \frac{d y}{d x} - y) y \sin \frac{y}{x} \\ \Rightarrow (x^{2} \cos \frac{y}{x} - x y \sin \frac{y}{x}) \frac{d y}{d x} = - x y \cos \frac{y}{x} - y^{2} \sin \frac{y}{x} \\ \Rightarrow \frac{d y}{d x} = \frac{x y \cos \frac{y}{x} + y^{2} \sin \frac{y}{x}}{x y \sin \frac{y}{x} - x^{2} \cos \frac{y}{x}} \\ \Rightarrow \frac{d y}{d x} = \frac{\frac{x y}{x^{2}} \cos \frac{y}{x} + \frac{y^{2}}{x^{2}} \sin \frac{y}{x}}{\frac{x y}{x^{2}} \sin \frac{y}{x} - \frac{x^{2}}{x^{2}} \cos \frac{y}{x}} \\ x \frac{d v}{d x} = \frac{v \cos v + v^{2} \sin v}{v \sin v - \cos v} - v = \frac{2 v \cos v}{v \sin v - \cos v} (on substituting y = vx) \\ \Rightarrow \int \frac{v \sin v - \cos v}{v \cos v} d v = 2 \int \frac{d x}{x} \\ \Rightarrow \int (Tan v - \frac{1}{v}) d v = 2 \ln x + \ln c ∣ \\ \Rightarrow \ln | \sec v | - \ln | ν | = 2 \ln x + \ln c \\ \Rightarrow \ln |\frac{\sec v}{v}| = \ln |c x^{2}| \\ \Rightarrow \frac{x}{y} \sec \frac{y}{x} = c x^{2} \end{array}$

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