`dy/dx = 6x^2` Use integration to find a general solution to the differential equation

Sunday, April 17, 2016

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={6}{{x}}^{{2}}$ Use integration to find a general solution to the differential equation

An ordinary differential equation (ODE) is differential equation for the derivative of a function of one variable. When an ODE is in a form of $\displaystyle{y}'={f{{\left({x},{y}\right)}}}$ , this is just a first order ordinary differential equation.

The $\displaystyle{y}'$ is the same as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ therefor first order ODE can written in a form of $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={f{{\left({x},{y}\right)}}}$

That is form of the given problem: (dy)/(dx) = 6x^2.

We may apply integration after we rearrange it in a form of variable separable differential equation: $\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$ .

By cross-multiplication, we can be rearrange the problem into: $\displaystyle{\left({\left.{d}{y}\right.}\right)}={6}{{x}}^{{2}}{\left.{d}{x}\right.}$ .

Apply direct integration on both sides:

$\displaystyle\int{\left({\left.{d}{y}\right.}\right)}=\int{6}{{x}}^{{2}}{\left.{d}{x}\right.}$ .

For the left side, we may apply basic integration property:

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

For the right side, we may apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

$\displaystyle\int{6}{{x}}^{{2}}{\left.{d}{x}\right.}={6}\int{{x}}^{{2}}{\left.{d}{x}\right.}$

Then apply Power Rule for integration: $\displaystyle\int{{u}}^{{n}}{d}{u}=\frac{{{u}}^{{{n}+{1}}}}{{{n}+{1}}}+{C}$

$\displaystyle{6}\int{{x}}^{{2}}{\left.{d}{x}\right.}={6}\cdot\frac{{{x}}^{{{2}+{1}}}}{{{2}+{1}}}$

$\displaystyle={6}\cdot\frac{{{x}}^{{3}}}{{3}}+{C}$

$\displaystyle={2}{{x}}^{{3}}+{C}$

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

$\displaystyle{y}={2}{{x}}^{{3}}+{C}$

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Sunday, April 17, 2016

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={6}{{x}}^{{2}}$ 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?

Sunday, April 17, 2016

Use integration to find a general solution to 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?

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}={6}{{x}}^{{2}}$ 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?