Question: "To find the equation: 2(2x3−y2)dx−xydy=0"

babeeb0oL

Answered question

2021-10-17

To find the equation:

2 (2 x^{3} - y^{2}) d x - x y d y = 0

oppturf

Skilled2021-10-18Added 94 answers

Step 1
Putting these values in (2), we get,

\frac{v d v}{4 - 3 v^{2}} = \frac{d x}{x}

\Rightarrow - \frac{d p}{6 p} = \frac{d x}{x}

\Rightarrow \frac{d p}{p} = - 6 (\frac{d x}{x})

Now integrating both sides of the above equation, we get,

\int \frac{d p}{p} = \int \frac{- 6 d x}{x}

\Rightarrow \int \frac{d p}{p} = - 6 \int \frac{d x}{x}

\Rightarrow \log (p) = - 6 \log (x) + \log (c)

[c is integrating constant]

\Rightarrow \log (p) = \log (c x^{(- 6)})

\Rightarrow p = \frac{c}{x^{6}}

\Rightarrow 4 - 3 v^{2} = \frac{c}{x^{6}}

[putting the value of p]

\Rightarrow 3 v^{2} = 4 - \frac{c}{x^{6}}

\Rightarrow v^{2} = \frac{1}{3} (\frac{4 x^{6} - c}{x^{6}})

\Rightarrow v = \pm \frac{1}{\sqrt{3}} (\frac{\sqrt{4 x^{6} - c}}{x^{3}})

\Rightarrow v = \pm \frac{1}{\sqrt{3} x^{3}} (\sqrt{4 x^{6} - c})

\Rightarrow \frac{y}{x} = \pm \frac{1}{\sqrt{3} x^{3}} (\sqrt{4 x^{6} - c})

[putting the value of v]

\Rightarrow y = \pm \frac{1}{\sqrt{3} x^{2}} (\sqrt{4 x^{6} - c})

Hence the required solution is,

y = \pm \frac{1}{\sqrt{3} x^{2}} (\sqrt{4 x^{6} - c})

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