`xy' = y` Find the general solution of the differential equation

Friday, April 17, 2009

$\displaystyle{x}{y}'={y}$ Find the general solution of the differential equation

An ordinary differential equation (ODE) has differential equation for a function with single variable. A first order ODE follows y' =f(x,y).

The $\displaystyle{y}'$ can be denoted as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ to be able to express in a variable separable differential equation: $\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$ .

To be able to follow this, we let $\displaystyle{y}'=\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ on the given first order ODE: $\displaystyle{x}{y}'={y}$ :

$\displaystyle{x}{y}'={y}$

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

Cross-multiply to rearrange it into:

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

Applying direct integration on both sides:

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

Apply basic integration formula for logarithm: $\displaystyle\int\frac{{{d}{u}}}{{u}}={\ln}{\left|{u}\right|}+{C}$ .

$\displaystyle{\ln}{\left|{y}\right|}={\ln}{\left|{x}\right|}+{C}$

$\displaystyle{y}={{e}}^{{{\ln}{\left|{x}\right|}+{C}}}$

$\displaystyle={C}{{e}}^{{\ln}}{\left|{x}\right|}$ since $\displaystyle{{e}}^{{C}}$ is a constant

$\displaystyle{y}={C}{x}$

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Friday, April 17, 2009

$\displaystyle{x}{y}'={y}$ Find the general solution of the differential equation

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Friday, April 17, 2009

Find the general solution of the differential equation

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{x}{y}'={y}$ Find the general solution of the differential equation

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?