`dy/dx = x+3` Solve the differential equation

Sunday, March 31, 2013

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}={x}+{3}$

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

$\displaystyle{\left.{d}{y}\right.}={\left({x}+{3}\right)}{\left.{d}{x}\right.}$

Integrating both sides, it result to

$\displaystyle\int{\left.{d}{y}\right.}=\int{\left({x}+{3}\right)}{\left.{d}{x}\right.}$

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

Isolating the y, it becomes

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

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

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

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

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