`dy/dx = 6 - y` Solve the differential equation

Sunday, September 11, 2011

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={6}-{y}$ Solve the differential equation

Apply direct integration both sides: $\displaystyle\int{N}{\left({y}\right)}{\left.{d}{y}\right.}=\int{M}{\left({x}\right)}{\left.{d}{x}\right.}$ to solve for the general solution of a differential equation.

For the given first order ODE: $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={6}-{y}$ it can be rearrange by cross-multiplication into:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{6}-{y}}}={\left.{d}{x}\right.}$

Apply direct integration on both sides: $\displaystyle\int\frac{{{\left.{d}{y}\right.}}}{{{6}-{y}}}=\int{\left.{d}{x}\right.}$

For the left side, we consider u-substitution by letting:

$\displaystyle{u}={6}-{y}$ then $\displaystyle{d}{u}=-{\left.{d}{y}\right.}$ or $\displaystyle-{d}{u}={\left.{d}{y}\right.}$

The integral becomes: $\displaystyle\int\frac{{{\left.{d}{y}\right.}}}{{{6}-{y}}}=\int\frac{{-{d}{u}}}{{{u}}}$

Applying basic integration formula for logarithm:

$\displaystyle\int\frac{{-{d}{u}}}{{{u}}}=-{\ln}{\left|{u}\right|}$

Plug-in $\displaystyle{u}={6}-{y}$ on $\displaystyle-{\ln}{\left|{u}\right|}$ , we get:

$\displaystyle\int\frac{{{\left.{d}{y}\right.}}}{{{6}-{y}}}=-{\ln}{\left|{6}-{y}\right|}$

For the right side, we apply the basic integration: $\displaystyle\int{\left.{d}{x}\right.}={x}+{C}$

Combing the results from both sides, we get the general solution of the differential equation as:

$\displaystyle-{\ln}{\left|{6}-{y}\right|}={x}+{C}$

$\displaystyle{y}={6}-{{e}}^{{{\left(-{x}-{C}\right)}}}$

$\displaystyle{y}={6}-{C}{{e}}^{{-{x}}}$

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Sunday, September 11, 2011

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={6}-{y}$ Solve 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?

Sunday, September 11, 2011

Solve 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?

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={6}-{y}$ Solve the differential equation

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