`yy' = -8cospix` Find the general solution of the differential equation

Saturday, June 13, 2015

$\displaystyle{y}{y}'=-{8}{\cos{\pi}}{x}$ Find the general solution of the differential equation

The general solution of a differential equation in a form of $\displaystyle{f{{\left({y}\right)}}}{y}'={f{{\left({x}\right)}}}$ can

be evaluated using direct integration.

We can denote $\displaystyle{y}'$ as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ then,

$\displaystyle{f{{\left({y}\right)}}}{y}'={f{{\left({x}\right)}}}$

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

Rearrange into : $\displaystyle{f{{\left({y}\right)}}}{\left({\left.{d}{y}\right.}\right)}={f{{\left({x}\right)}}}{\left.{d}{x}\right.}$

To be able to apply direct integration : $\displaystyle\int{f{{\left({y}\right)}}}{\left({\left.{d}{y}\right.}\right)}=\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}.$

Applying this to the given problem: $\displaystyle{y}{y}'=-{8}{\cos{{\left(\pi{x}\right)}}}$ , we get:

$\displaystyle{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=-{8}{\cos{{\left(\pi{x}\right)}}}$

$\displaystyle{y}{\left({\left.{d}{y}\right.}\right)}=-{8}{\cos{{\left(\pi{x}\right)}}}{\left.{d}{x}\right.}$

$\displaystyle\int{y}{\left({\left.{d}{y}\right.}\right)}=\int-{8}{\cos{{\left(\pi{x}\right)}}}{\left.{d}{x}\right.}$

For the integration on the left side, we apply Power Rule integration: int u^n $\displaystyle{d}{u}=\frac{{{u}}^{{{n}+{1}}}}{{{n}+{1}}}$ on int $\displaystyle{y}{\left.{d}{y}\right.}$ .

$\displaystyle\int{y}{\left.{d}{y}\right.}=\frac{{{y}}^{{{1}+{1}}}}{{{1}+{1}}}$

$\displaystyle=\frac{{{y}}^{{2}}}{{2}}$

For the integration on the right side, we apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ and basic integration formula for cosine function: $\displaystyle\int{\cos{{\left({u}\right)}}}{d}{u}={\sin{{\left({u}\right)}}}+{C}$

$\displaystyle\int-{8}{\cos{{\left(\pi{x}\right)}}}{\left.{d}{x}\right.}=-{8}\int{\cos{{\left(\pi{x}\right)}}}{\left.{d}{x}\right.}$

Let $\displaystyle{u}=\pi{x}$ then $\displaystyle{d}{u}=\pi{\left.{d}{x}\right.}$ or $\displaystyle\frac{{{d}{u}}}{\pi}={\left.{d}{x}\right.}.$

Then the integral becomes:

$\displaystyle-{8}\int{\cos{{\left(\pi{x}\right)}}}{\left.{d}{x}\right.}=-{8}\int{\cos{{\left({u}\right)}}}\cdot\frac{{{d}{u}}}{\pi}$

$\displaystyle=-\frac{{8}}{\pi}\int{\cos{{\left({u}\right)}}}{d}{u}$

$\displaystyle=-\frac{{8}}{\pi}\cdot{\sin{{\left({u}\right)}}}+{C}$

Plug-in $\displaystyle{u}=\pi{x}$ in $\displaystyle-\frac{{8}}{\pi}\cdot{\sin{{\left({u}\right)}}}+{C}$ , we get:

$\displaystyle-{8}\int{\cos{{\left(\pi{x}\right)}}}{\left.{d}{x}\right.}=-\frac{{8}}{\pi}\cdot{\sin{{\left(\pi{x}\right)}}}+{C}$

Combing the results, we get the general solution for differential equation $\displaystyle{\left({y}{y}'=-{8}{\cos{{\left(\pi{x}\right)}}}\right)}$ as:

$\displaystyle\frac{{{y}}^{{2}}}{{2}}=-\frac{{8}}{\pi}\cdot{\sin{{\left(\pi{x}\right)}}}+{C}$

$\displaystyle{2}\cdot{\left[\frac{{{y}}^{{2}}}{{2}}\right]}={2}\cdot{\left[-\frac{{8}}{\pi}\cdot{\sin{{\left(\pi{x}\right)}}}\right]}+{C}$

$\displaystyle{{y}}^{{2}}=-\frac{{16}}{\pi}\cdot{\sin{{\left(\pi{x}\right)}}}+{C}$

The general solution: $\displaystyle{{y}}^{{2}}=-\frac{{16}}{\pi}{\sin{{\left(\pi{x}\right)}}}+{C}$ can be expressed as:

$\displaystyle{y}=\pm\sqrt{{-\frac{{16}}{\pi}{\sin{{\left(\pi{x}\right)}}}+{C}}}$ .

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Saturday, June 13, 2015

$\displaystyle{y}{y}'=-{8}{\cos{\pi}}{x}$ 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?

Saturday, June 13, 2015

Find the general solution of the differential equation

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

$\displaystyle{y}{y}'=-{8}{\cos{\pi}}{x}$ 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?