`dy/dx = (6-x^2)/ (2y^3)` Find the general solution of the differential equation

Saturday, October 8, 2011

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}=\frac{{{6}-{{x}}^{{2}}}}{{{2}{{y}}^{{3}}}}$ Find the general solution of the differential equation

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{{6}-{{x}}^{{2}}}}{{{2}{{y}}^{{3}}}}$

This differential equation is separable since it can be rewritten in the form

$\displaystyle{N}{\left({y}\right)}{\left.{d}{y}\right.}={M}{\left({x}\right)}{\left.{d}{x}\right.}$

So separating the variables, the equation becomes

$\displaystyle{2}{{y}}^{{3}}{\left.{d}{y}\right.}={\left({6}-{{x}}^{{2}}\right)}{\left.{d}{x}\right.}$

Integrating both sides, it result to

$\displaystyle\int{2}{{y}}^{{3}}{\left.{d}{y}\right.}=\int{\left({6}-{{x}}^{{2}}\right)}{\left.{d}{x}\right.}$

$\displaystyle{2}\cdot\frac{{{y}}^{{4}}}{{4}}+{C}_{{1}}={6}{x}-\frac{{{x}}^{{3}}}{{3}}+{C}_{{2}}$

$\displaystyle\frac{{{y}}^{{4}}}{{2}}+{C}_{{1}}=-\frac{{{x}}^{{3}}}{{3}}+{6}{x}+{C}_{{2}}$

Since C1 and C2 represent any number, it can be expressed as a single constant C.

$\displaystyle\frac{{{y}}^{{4}}}{{2}}=-\frac{{{x}}^{{3}}}{{3}}+{6}{x}+{C}$

Therefore, the general solution of the given differential equation is $\displaystyle\frac{{{y}}^{{4}}}{{2}}=-\frac{{{x}}^{{3}}}{{3}}+{6}{x}+{C}$ .

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Saturday, October 8, 2011

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}=\frac{{{6}-{{x}}^{{2}}}}{{{2}{{y}}^{{3}}}}$ 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, October 8, 2011

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?

$\displaystyle\frac{{\left.{d}{y}}}{{\left.{d}{x}}}=\frac{{{6}-{{x}}^{{2}}}}{{{2}{{y}}^{{3}}}}$ 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?