First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation, $y d x + (x + x^{2} y) d y = 0$ is

Moderate

A

$\log y = c x$
B

$\frac{1}{x y} + \log y = c$
C

$- \frac{1}{x y} + \log y = c$
D

$- \frac{1}{x y} + \log x = c$

Solution

Given differential equation is
$\begin{array}{l} y d x + (x + x^{2} y) d y = 0 or y d x + x d y + x^{2} y d y = 0 \\ \Rightarrow \frac{y d x + x d y}{x^{2} y} + d y = 0 \\ \Rightarrow \frac{y d x + x d y}{x^{2} y^{2}} + \frac{d y}{y} = 0 \\ \Rightarrow \frac{d y}{y} = - (x y)^{- 2} d (x y) \\ \Rightarrow d (\ln y) = d ((x y)^{- 1}) \\ Integrate we get \Rightarrow \ln y = \frac{1}{x y} + c \end{array}$

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