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- Feb 15th 2016, 06:52 AM #1

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- Feb 15th 2016, 07:44 AM #2

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- Feb 15th 2016, 07:47 AM #3

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- Feb 15th 2016, 07:58 AM #4

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- Feb 15th 2016, 08:02 AM #5

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- Feb 15th 2016, 04:44 PM #6
## Re: dy/ dx + (y/x) = 2(x^2)

This equation is first order linear, just use an integrating factor...

$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{1}{x}\,y &= 2\,x^2 \end{align*}$

Multiply both sides by the integrating factor $\displaystyle \begin{align*} \mathrm{e}^{\int{\frac{1}{x}\,\mathrm{d}x}} = \mathrm{e}^{\ln{(x)}} = x \end{align*}$ and the equation becomes

$\displaystyle \begin{align*} x\,\frac{\mathrm{d}y}{\mathrm{d}x} + y &= 2\,x^3 \\ \frac{\mathrm{d}}{\mathrm{d}x} \, \left( x\,y \right) &= 2\,x^3 \\ x\,y &= \int{ 2\,x^3\,\mathrm{d}x} \\ x\,y &= \frac{x^4}{2} + C \\ y &= \frac{x^3}{2} + \frac{C}{x} \end{align*}$