`y' = 5x/y` Solve the differential equation

Thursday, October 10, 2013

$\displaystyle{y}'={5}\frac{{x}}{{y}}$ Solve the differential equation

The general solution of a differential equation in a form of $\displaystyle{y}'={f{{\left({x},{y}\right)}}}$ can be evaluated using direct integration. The derivative of y denoted as $\displaystyle{y}'$ can be written as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ then $\displaystyle{y}'={f{{\left({x}\right)}}}$ can be expressed as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={f{{\left({x}\right)}}}$ .

For the problem: $\displaystyle{y}'={5}\frac{{x}}{{y}}$ , we let $\displaystyle{y}'=\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ to set it up as:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={5}\frac{{x}}{{y}}$

Cross-multiply $\displaystyle{\left.{d}{x}\right.}$ to the right side:

$\displaystyle{\left({\left.{d}{y}\right.}\right)}={5}\frac{{x}}{{y}}{\left.{d}{x}\right.}$

Cross-multiply y to the left side:

$\displaystyle{y}{\left.{d}{y}\right.}={5}{x}{\left.{d}{x}\right.}$

Apply direct integration on both sides:

$\displaystyle\int{y}{\left.{d}{y}\right.}=\int{5}{x}{\left.{d}{x}\right.}$

Apply basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ on the right side.