First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $x \frac{d y}{d x} + y = x^{3} y^{6}$ is

Difficult

A

$y^{- 5} x^{- 5} = \frac{5}{2} x^{- 2} + c$
B

$y^{5} x^{5} = \frac{5}{2} x^{- 2} + c$
C

$y^{- 5} x^{- 5} = \frac{5}{2} x^{2} + c$
D

none of these

Solution

Given differential equation is
$\begin{array}{l} x \frac{d y}{d x} + y = x^{3} y^{6} \\ \Rightarrow x d y + y d x = x^{3} y^{6} d x \Rightarrow x d y + y d x = \frac{x^{6} y^{6}}{x^{3}} d x \\ \Rightarrow \frac{d (x y)}{{(x y)}^{6}} = \frac{d x}{x^{3}} \\ i n t e g r a t e o n b o t h s i d e s \\ \Rightarrow - \frac{1}{5} (x y)^{- 5} = - \frac{x^{- 2}}{2} + C_{1} \\ \Rightarrow y^{- 5} x^{- 5} = \frac{5}{2} x^{- 2} - 5 C_{1} \Rightarrow y^{- 5} x^{- 5} = \frac{5}{2} x^{- 2} + c \end{array}$

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