`dy/dx = xsqrt(x-6)` Use integration to find a general solution to the differential equation

Friday, January 24, 2014

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={x}\sqrt{{{x}-{6}}}$ Use integration to find a general solution to 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 $\displaystyle{y}'={f{{\left({x},{y}\right)}}}$ can be expressed as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={f{{\left({x},{y}\right)}}}$ .

That is form of the given problem: $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={x}\sqrt{{{x}-{6}}}$ .

We may apply the variable separable differential in which we follow $\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$ .

Cross-multiply $\displaystyle{\left.{d}{x}\right.}$ to the right side: $\displaystyle{\left.{d}{y}\right.}={x}\sqrt{{{x}-{6}}}{\left.{d}{x}\right.}$ .

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

For the left side, we apply basic integration property:

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

For the right side, we may apply u-substitution by letting: $\displaystyle{u}={x}-{6}$ or $\displaystyle{x}={u}+{6}$ then $\displaystyle{\left.{d}{x}\right.}={d}{u}$ .

$\displaystyle\int{x}\sqrt{{{x}-{6}}}{\left.{d}{x}\right.}=\int{\left({u}+{6}\right)}\sqrt{{{u}}}{d}{u}$

$\displaystyle=\int{\left({u}+{6}\right)}{{u}}^{{\frac{{1}}{{2}}}}{d}{u}$

$\displaystyle=\int{\left({{u}}^{{\frac{{3}}{{2}}}}+{6}{{u}}^{{\frac{{1}}{{2}}}}\right)}{d}{u}$

Apply the basic integration property: $\displaystyle\int{\left({u}+{v}\right)}{\left.{d}{x}\right.}=\int{\left({u}\right)}{\left.{d}{x}\right.}+\int{\left({v}\right)}{\left.{d}{x}\right.}$ .

$\displaystyle\int{{u}}^{{\frac{{3}}{{2}}}}{d}{u}+\int{6}{{u}}^{{\frac{{1}}{{2}}}}{d}{u}$

Apply the Power Rule for integration : $\displaystyle\int{{x}}^{{n}}=\frac{{{x}}^{{{n}+{1}}}}{{{n}+{1}}}+{C}$ .

$\displaystyle\int{{u}}^{{\frac{{3}}{{2}}}}{d}{u}+\int{6}{{u}}^{{\frac{{1}}{{2}}}}{d}{u}=\frac{{{u}}^{{{\left(\frac{{3}}{{2}}+{1}\right)}}}}{{\frac{{3}}{{2}}+{1}}}+{6}\frac{{{u}}^{{{\left(\frac{{1}}{{2}}+{1}\right)}}}}{{\frac{{1}}{{2}}+{1}}}+{C}$

$\displaystyle=\frac{{{u}}^{{\frac{{5}}{{2}}}}}{{{\left(\frac{{5}}{{2}}\right)}}}+{6}\frac{{{u}}^{{\frac{{3}}{{2}}}}}{{{\left(\frac{{3}}{{2}}\right)}}}+{C}$

$\displaystyle={{u}}^{{\frac{{5}}{{2}}}}\cdot{\left(\frac{{2}}{{5}}\right)}+{6}{{u}}^{{\frac{{3}}{{2}}}}\cdot{\left(\frac{{2}}{{3}}\right)}+{C}$

$\displaystyle=\frac{{{2}{{u}}^{{\frac{{5}}{{2}}}}}}{{5}}+{4}{{u}}^{{\frac{{3}}{{2}}}}+{C}$

Plug-in $\displaystyle{u}={x}-{6}$ , we get:

$\displaystyle\int{x}\sqrt{{{x}-{6}}}{\left.{d}{x}\right.}=\frac{{{2}{{\left({x}-{6}\right)}}^{{\frac{{5}}{{2}}}}}}{{5}}+{4}{{\left({x}-{6}\right)}}^{{\frac{{3}}{{2}}}}+{C}$

Combining the results, we get the general solution for the differential equation:

$\displaystyle{y}=\frac{{{2}{{\left({x}-{6}\right)}}^{{\frac{{5}}{{2}}}}}}{{5}}+{4}{{\left({x}-{6}\right)}}^{{\frac{{3}}{{2}}}}+{C}$

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Friday, January 24, 2014

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={x}\sqrt{{{x}-{6}}}$ Use integration to find a general solution to 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, January 24, 2014

Use integration to find a general solution to 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?

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={x}\sqrt{{{x}-{6}}}$ Use integration to find a general solution to the differential equation

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