Separation of variables

A symbolic algorithm for the maxima CAS Nijso Beishuizen

We discuss a simple but effective implementation of a method to separate variables. Separation of variables is a well-known and much-used technique to solve ordinary and partial differential equations.

An expression $F(x,y)$ with the 2 independent variables $x$ and $y$ is separable when $\frac{\partial F}{\partial x}/F = \frac{F_x}{F}$ depends only on $x$, see 1, 2. This is easy to see because if $F$ is separable, it can be written as $F=f(x)\cdot R(y)$. If on both sides we take the derivative with respect to $x$ and divide by $F$ we obtain: $\frac{F_x}{F} = \frac{f_x}{f}$. Since we know that the right-hand side is a function of only x, the left-hand side must be as well. The separated function $f(x)$ can then be found by solving the differential equation:

$|f(x)| = e^{\int(F_x/Fdx)}$ (1)

The function $g(y)$ can be found in a similar way, e.g. by computing:

$|g(y)| = e^{\int(F_y/Fdy)}$, (2)

or by using $g(y) = F(x,y)/f(x)$. Note that the latter approach might be computationally more costly because we have to properly factor out the x-dependency. With the above aproach eqs. (1)-(2), we lose any constant factors, e.g. when $F(x,y)=C \cdot f(x) \cdot g(y)$, we have lost $C$. We can find $C$ during the evaluation of our results: we check that $C = \frac{F(x,y)}{f(x)\cdot g(y)}$ is free of $x$ and $y$.

The maxima implementation for separation of variables has been implemented as follows:

Function and variable index

Function: separable(expr,x,y,[options])

If the input expression is a separable function in the independent variables $x$ and $y$, e.g. the expression can be written as $C\cdot f(x)\cdot g(y)$, with $C$ a constant independent of $x,y$, then separable(expr,x,y) returns a list of the form $[f(x), C\cdot g(y)]$, or false when the expression is not separable. When the optional input option 'splitConstant=true is set, the constant factor will be separated as well and a list $[f(x),g(y),C]$ will be returned.

Function: isSeparable(expr,x,y)

If the input expression can be written as a separable expression $f(x)\cdot g(y)$, return true, otherwise return false.

Function: constant_factors(expr,varlist)

If the input expression can be written as $C\cdot F(x,y)$, with $C$ an expression free of the variables in the list varlist, constant_factors will return the constant, or 1.

Example

For some expressions is it not immediately clear from inspection that they are separable. For instance the expression

$e^{x^2+y^2}(\cos(x+y) + \cos(x-y))$

can be rewritten as the product of separable functions:

$e^{x^2} \cdot e^{y^2} \cdot 2\cos(x) \cos(y)$

The maxima implementation of the above algorithm recognizes this without relying on pattern matching to recognize this separation.

The constant factor is always added to the second term $g(y)$. We can also separate the constant by using the optional command 'splitConstant=true. The constant factor is then added to the list.

We can also call the routine responsible for the separation of the constant factor independently:

constant_factors also tries to find a shared minus sign. If all terms in an expression of the form a+b+c+d have a minus sign, e.g. expr: -a-b-c-d, then the minus sign will also be factored out.

Sometimes we just want to know if an expression is separable or not. We can then skip some internal computations.

Separation of variables for an ODE

We can use separation of variables to solve ordinary differential equations. For the general separable ode:

Bibliography

[1] C.P. Viazminsky, On separation of variables, arXiv:math/0210167 (https://arxiv.org/pdf/math/0210167.pdf)

[2] J.A. Cid, A simple method to find out when and ordinary differential equation is separable, International Journal of Mathematical Education in Science and Technology Vol. 40 , Iss. 5,2009 (http://www4.ujaen.es/~angelcid/Archivos/Papers/IJMEST.pdf)