First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $y^{2} d y = x (x d y - y d x) e^{x / y}$ is

Moderate

A

$e^{x y} (x y - 1) + \log y = c$
B

$e^{x / y} (x / y - 1) + \log y = c$
C

$e^{y / x} (y / x - 1) + \log x = c$
D

none

Solution

The given equation can be re written as
$\begin{array}{l} \Rightarrow (y^{2} - x^{2} e^{x / y}) d y = - x y e^{x / y} d x \\ \Rightarrow \frac{d x}{d y} = \frac{x^{2} e^{x / y} - y^{2}}{x y e^{x / y}} = \frac{x}{y} - \frac{1}{\frac{x}{y} e^{x / y}} \\ Put x = v y then \Rightarrow \frac{d x}{d y} = v + y \frac{d v}{d y} \\ \Rightarrow v + y \frac{d v}{d y} = v - \frac{1}{v e^{v}} \end{array}$
$\begin{array}{l} \Rightarrow \int v e^{v} d v = - \int \frac{d y}{y} \\ \Rightarrow e^{v} (v - 1) = - \ln | y | + c \\ \Rightarrow e^{x / y} (\frac{x}{y} - 1) + \ln y = c \end{array}$

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