`dy/dx = 5 - 8x` Solve the differential equation

Friday, August 7, 2009

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}={5}-{8}{x}$

This differential equation is separable since it has a form

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

And, it can be re-written as

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

So separating the variables, the equation becomes

$\displaystyle{\left.{d}{y}\right.}={\left({5}-{8}{x}\right)}{\left.{d}{x}\right.}$

Integrating both sides, it result to

$\displaystyle\int{\left.{d}{y}\right.}=\int{\left({5}-{8}{x}\right)}{\left.{d}{x}\right.}$

$\displaystyle{y}+{C}_{{1}}={5}{x}-\frac{{{8}{{x}}^{{2}}}}{{2}}+{C}_{{2}}$

$\displaystyle{y}+{C}_{{1}}={5}{x}-{4}{{x}}^{{2}}+{C}_{{2}}$

Isolating the y, it becomes

$\displaystyle{y}={5}{x}-{4}{{x}}^{{2}}+{C}_{{2}}-{C}_{{1}}$

Since C2 and C1 are constants, it can be expressed as a single constant C.

$\displaystyle{y}={5}{x}-{4}{{x}}^{{2}}+{C}$

$\displaystyle{y}=-{4}{{x}}^{{2}}+{5}{x}+{C}$

Therefore, the general solution of the given differential equation is $\displaystyle{y}=-{4}{{x}}^{{2}}+{5}{x}+{C}$ .

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