First slide

Methods of solving first order first degree differential equations

Question

The solution of $y^{5} x + y - x \frac{d y}{d x} = 0$ is

Difficult

A

$\frac{x^{4}}{4} + \frac{1}{5} . \frac{x^{5}}{y^{5}} = C$
B

$\frac{x^{5}}{5} + \frac{1}{4} \frac{x^{4}}{y^{4}} = C$
C

$\frac{x^{5}}{y^{5}} + \frac{1}{5} \frac{x^{4}}{y^{4}} = C$
D

none of these

Solution

$\begin{array}{l} y^{5} x d x + y d x - x d y = 0 \\ \Rightarrow y^{3} x d x + \frac{y d x - x d y}{y^{2}} = 0 \\ y^{3} x d x + d (\frac{x}{y}) = 0 \\ \Rightarrow x d x + \frac{1}{y^{3}} d (\frac{x}{y}) = 0 \\ \Rightarrow x^{4} d x + {(\frac{x}{y})}^{3} d (\frac{x}{y}) = 0 \\ Integrate we get \frac{x^{5}}{5} + \frac{(\frac{x}{y})^{4}}{4} = C \end{array}$

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