Separable Equations

Non-linear differential equations come in many forms. One of these forms is separable equations. A differential equation that is separable will have several properties which can be exploited to find a solution.

A separable equation is a differential equation of the following form:

$\displaystyle{N(y)\frac{dy}{dx}=M(x)}$

The original equation is separable if this differential equation can be expressed as:

$f(x)dx + g(y)dy = 0$

 where $f(x)$ is in terms of only $x$ and $g(y)$ is in terms of only $y$. This is the easiest variety of differential equation to solve. Integrating such an equation yields:

$\int f(x)dx + \int g(y)dy = c$

where $c$ is the standard arbitrary constant.

To separate the equations means to move all the $x$ terms and $y$s terms to the opposite sides of the equation.

A general approach to solving separable equations is as follows:

• Multiply and divide to get rid of any fractions.
• Combine any terms involving the same differential into one term.
• Integrate each component on its own, and don't forget to add constants to equations after integrating. This ensures that the solution is of the general form.
• Finally, simplify the expression (i.e., combine all possible terms, rewrite any logarithmic terms in exponent form, and express any arbitrary constants in the most simple terms possible).

After simplifying you will have the general form of the equation. A particular solution to the equation will depend on the choice of the arbitrary constants you obtained when integrating.

For example, consider the time-independent Schrödinger equation :

$\left( -\bigtriangledown^{2} + V(x) \right) \cdot \psi (x) = E\psi (x)$

If the function $V (x) $ in three dimensions is of the form

$V(x_{1}, x_{2}, x_{3}) = V_{1}(x_{1})+ V_{2}(x_{2}) + V_{3}(x_{3})$

then it turns out that the problem can be separated into three one-dimensional ordinary differential equations for functions: $\psi_{1} (x_{1})$, $\psi_{2} (x_{2})$, $\psi_{3} (x_{3})$ .

The final solution can be written as follows:

$\psi(x) = \psi_{1} (x_{1})\cdot \psi_{2} (x_{2})\cdot \psi_{3} (x_{3})$

Non-Relativistic Schrödinger Equation

A wave function which satisfies the non-relativistic Schrödinger equation with $V=0$. This corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here.