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Variable Separable Method

dydx = x2 x -...

Question

$\frac{d y}{d x} = x^{2} (x - 2)$
Given $y = 2$ ; when $x = 0$ .

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Solution

We have,

$\frac{d y}{d x} = x^{2} (x - 2)$

$\frac{d y}{d x} = x^{3} - 2 x^{2}$

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

On taking integral both sides, we get

$\int d y = \int (x^{3} - 2 x^{2}) d x$

$y = \frac{x^{4}}{4} - \frac{2 x^{3}}{3} + c$

At $x = 0, y = 2$

So,

$2 = 0 - 0 + c$

$c = 2$

Therefore, the general solution

$y = \frac{x^{4}}{4} - \frac{2 x^{3}}{3} + 2$

Hence, this is the answer.

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