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Find the general solution of the differential equations:{({x}+

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Find the general solution of the differential equations: $(x + 3 y^{2}) \frac{d x}{d y} = y (y > 0)$

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Find the general solution of the differential equations:

(x + 3 y^{2}) \frac{d x}{d y} = y (y > 0)

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Text solutionVerified

Solution: Here, given equation is,

(x + 3 y^{2}) \frac{d y}{d x} = y

\Rightarrow \frac{d x}{d y} = \frac{x + 3 y ^{2}}{y}

\Rightarrow \frac{d x}{d y} - \frac{x}{y} = 3 y

Comparing it with

\frac{d x}{d y} + P x = Q

P = - \frac{1}{y} and Q = 3 y

Integrating factor,

I . F . = e^{\int P d y}

I . F . = e^{\int - \frac{1}{y} d y} = e^{- l n y} = y^{- l n e} = y - 1 = \frac{1}{y}

Now, general solution will be,

x (I . F .) = \int (I . F .) Q d y

\Rightarrow \frac{x}{y} = \int (\frac{1}{y}) (3 y) d y

\Rightarrow \frac{x}{y} = 3 y + c

\Rightarrow x = 3 y^{2} + c y

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Question Text	Find the general solution of the differential equations: $(x + 3 y^{2}) \frac{d x}{d y} = y (y > 0)$
Answer Type	Text solution:1
Upvotes	150