Difference between revisions of "Separation of variables"
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| − | '''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== |
| + | 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> | ||
| + | |||
==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>. | ||
| + | |||
| + | |||
| + | ==References== | ||
*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 | ||
| + | [[Category:Algebra]] | ||
Latest revision as of 19:09, July 13, 2016
Separation of variables is a technique of solving a specific type of differential equations.
Definition
An ordainary differential equation is called separable if the following criteria can be satisfied through rearranging the equation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(y\right)\,dy=g\left(x\right)\,dx}
Example
To solve
We can rearrange the equation into
Integrating:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int y\,dy=\int \frac{1}{x}\,dx}
Which gives
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}y^2+C_1=\ln\left(x\right)+C_2}
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_2} are arbitaray constants, we can group them together and give
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}y^2=\ln\left(x\right)+C}
The answer is usually leave at the form described above instead of isolating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} .
References
- D. Lomen and D. Lovelock, Differential Equations Graphics. Model. Data., John Wiley and Sons, Toronto, 1999.
- Separation of variables on Wolfram Mathworld