`dy/dx = x/y` Find the general solution of the differential equation

Wednesday, March 27, 2013

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{x}}{{y}}$

This differential equation is separable since it can be re-written in the form

$\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$

So separating the variables, the equation becomes

$\displaystyle{y}{\left.{d}{y}\right.}={x}{\left.{d}{x}\right.}$

Integrating both sides, it result to

$\displaystyle\int{y}{\left.{d}{y}\right.}=\int{x}{\left.{d}{x}\right.}$

$\displaystyle\frac{{{y}}^{{2}}}{{2}}+{C}_{{1}}=\frac{{{x}}^{{2}}}{{2}}+{C}_{{2}}$

Isolating the y, it becomes

$\displaystyle\frac{{{y}}^{{2}}}{{2}}=\frac{{{x}}^{{2}}}{{2}}+{C}_{{2}}-{C}_{{1}}$

$\displaystyle{{y}}^{{2}}={{x}}^{{2}}+{2}{C}_{{2}}-{2}{C}_{{1}}$

$\displaystyle{y}=\pm\sqrt{{{{x}}^{{2}}+{2}{C}_{{2}}-{2}{C}_{{1}}}}$

Since C2 and C1 represents any number, it can be expressed as a single constant C.

$\displaystyle{y}=\pm\sqrt{{{{x}}^{{2}}+{C}}}$

Therefore, the general solution of the given differential equation is $\displaystyle{y}=\pm\sqrt{{{{x}}^{{2}}+{C}}}$ .

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