First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $x \frac{d y}{d x} = y + x e^{y / x}$ is

Moderate

A

$e^{- y / x} + \log x = c$
B

$e^{x / y} + \log y = c$
C

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

none

Solution

The given equation can be written as
$\begin{array}{l} \Rightarrow \frac{d y}{d x} = \frac{y}{x} + e^{y / x} p u t y = v x t h e n \frac{d y}{d x} = v + x \frac{d v}{d x} \\ v + x \frac{d v}{d x} = v + e^{v} \\ \Rightarrow \int e^{- v} d v = \int \frac{d x}{x} (b y v a r i a b l e s s e p a r a b l e) \\ \Rightarrow - e^{- v} + c = \log x \\ \Rightarrow e^{- y / x} + \log x = c \end{array}$

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