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Line 1: − + − A differential equation is called '''separable''' if the following criteria can be satisfied through rearranging the equation:+ − − <!-- − + − -->+ + + + + + + + +
added example
'''Separation of variables''' is a technique of solving [[differential equation]]s.
'''Separation of variables''' is a technique of solving a specific type of [[differential equation]]s.
==Definition==
==Definition==
An ordainary differential equation is called '''separable''' if the following criteria can be satisfied through rearranging the equation:
:<math>f\left(y\right)\,dy=g\left(x\right)\,dx</math>
:<math>f\left(y\right)\,dy=g\left(x\right)\,dx</math>
==Example==
==Example==
To solve
To solve
:<math>x\frac{dy}{dx}y=1</math>
:<math>x\frac{dy}{dx}y=1</math>
We can rearrange the equation into
We can rearrange the equation into
:<math>y\,dy=frac{1}{x}\,dx</math>
:<math>y\,dy=\frac{1}{x}\,dx</math>
Integrating:
:<math>\int y\,dy=\int \frac{1}{x}\,dx</math>
Which gives
:<math>\frac{1}{2}y^2+C_1=\ln\left(x\right)+C_2</math>
Since <math>C_1</math> and <math>C_2</math> are arbitaray constants, we can group them together and give
:<math>\frac{1}{2}y^2=\ln\left(x\right)+C</math>
The answer is usually leave at the form described above instead of isolating <math>y</math>.
==Reference==
==Reference==
*D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
*D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
*[http://mathworld.wolfram.com/SeparationofVariables.html Separation of variables] on Wolfram Mathworld
*[http://mathworld.wolfram.com/SeparationofVariables.html Separation of variables] on Wolfram Mathworld