Separation of variables

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:

$f\left(y\right)\,dy=g\left(x\right)\,dx$

Example

To solve

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 x\frac{dy}{dx}y=1}

We can rearrange the equation into

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\,dy=\frac{1}{x}\,dx}

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

${\frac {1}{2}}y^{2}=\ln \left(x\right)+C$

The answer is usually leave at the form described above instead of isolating $y$.

Reference

• D. Lomen and D. Lovelock, Differential Equations Graphics. Model. Data., John Wiley and Sons, Toronto, 1999.
• Separation of variables on Wolfram Mathworld