First slide

Methods of solving first order first degree differential equations

Question

Solution of the differential equation $y - x \frac{d y}{d x} = a (y^{2} + \frac{d y}{d x})$

Moderate

A

$(x + a) (1 - a y) = c y$
B

$(x - a) (1 - a y) = c y$
C

$(x - a) (1 - a y) = c x$
D

none

Solution

$\begin{array}{l} We have y - x \frac{d y}{d x} = a (y^{2} + \frac{d y}{d x}) \\ \Rightarrow y d x - x d y = a y^{2} d x + a d y \\ \Rightarrow y (1 - a y) d x = (x + a) d y \\ \Rightarrow \frac{d x}{x + a} - \frac{d y}{y (1 - a y)} = 0 (since \frac{1}{y (1 - a y)} = \frac{1}{y} + \frac{a}{1 - a y}) \\ I n t e g r a t i n g, w e g e t \\ \log (x + a) - \log y - a (\frac{\log (1 - a y)}{- a}) = \log c \\ \log (x + a) - \log y + \log (1 - a y) = \log c \\ \Rightarrow \log \frac{(a + x) (1 - a y)}{y} = \log c \\ \Rightarrow (x + a) (1 - a y) = c y \end{array}$

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