Partial differential equations
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Given a partial differential equation of a function
of n variables, it is sometimes useful to guess solution of the form
or
which turns the partial differential equation (PDE) into a set of ODEs. Usually, each independent variable creates a separation constant that cannot be determined only from the equation itself.
When such a technique works, it is called a separable partial differential equation.
Suppose F(x, y, z) and the following PDE:
We shall guess
thus making the equation (1) to
(since ).
Now, since X'(x) is dependent only on x and Y'(y) is dependent only on y (so on for Z'(z)) and that the equation (1) is true for every x, y, z it is clear that each one of the term is constant. More precisely,
where the constants c1, c2, c3 satisfy
Eq. (3) is actually a set of three ODEs. In this case they are trivial and can be solved by simple integration, giving:
where the integration constant c4 is determined by initial conditions.
Consider the differential equation
First we seek solutions of the form
Most solutions are not of that form, but other solutions are sums of (generally infinitely many) solutions of that form.
Substituting,
Divide throughout by X(x)
and then by Y(y)
Now X′′(x)/X(x) is a function of x only, and (Y′′(y)+λY(y))/Y(y) is a function of y only, so for their sum to be equal to zero for all x and y, they must both be constant. Thus,
where k is the separation constant. This splits up into ordinary differential equations
and
which we can solve accordingly. If the equation as posed originally was a boundary value problem, one would use the given boundary values. See that article for an example which uses boundary values.