Separation of variables

The method of separation of variables, separation of variables, separation or separation of variables method is a method in the theory of ordinary differential equations. It can be used separable first order differential equations to solve. These are differential equations in which the first derivative of a product of a only and only dependent function is: The term " separation of variables " goes back to Johann Bernoulli, who used it in 1694 in a letter to Gottfried Wilhelm Leibniz.

A similar procedure for certain partial differential equations is the separation approach.

Solution of the initial value problem

We investigate the initial value problem

For continuous ( real) functions. If so this initial value problem is solved by a constant function. This solution need not be unique under the specified conditions.

Wording of the sentence

Let with. Then:

  • There is a comprehensive open interval with for all. Then the map is well defined and strictly monotone on.
  • Let and be as above. Then is the unique solution of the initial value problem

The solution of the initial value problem is in this case the solution of the equation

Note that in the case of the concrete form of the separate variables actually present at local uniqueness, need to meet though, and no local Lipschitz condition.

Evidence

Here and continuous, there is a comprehensive open interval, such that for all. In particular, on the same sign, so that is on well-defined and strictly monotone. 0 is a comprehensive open interval. So there is a comprehensive open interval, such that for all.

On well-defined, and is due for all

On. When deriving the chain rule and the reverse control were used. Is natural. This proves the existence of a solution of the given initial value problem.

For the uniqueness, assume that any solution of the initial value problem is on. It is now shown that applies to; the uniqueness of the left is analogous.

Assuming the uniqueness of the right would be violated. Because of the continuity of and there is with so

Is true, but for which the statement

For each with is wrong. In the following, we show that there still is a positive, is true for the above statement, which implies the desired contradiction.

Because there are with, such that for all. In particular, in a well- defined, and it is

This implies, that for all, which is consistent with the definition of. This provides the contradiction to the assumption of Nichteindeutigeit.

Example

Wanted is the solution of the initial value problem

This is a differential equation with separated variables:

So Set

The inverse function is

So the solution of the initial value problem is given by

Graphic

Clearly stating the set of the separation of variables that the following procedure is allowed, ie correct results (although the differentials and really only symbols are, strictly speaking, with which one can not count ):

  • Write the derivation consistently as.
  • Return all terms in which one occurs - including the - on the right, and everyone else - including the - on the left side, using ordinary fractions.
  • It should then be left in the numerator and the right of the counter a stand.
  • Just set before both sides an integral and integrate.
  • Determine integration constant using the initial condition.

The invoice for the above example would then proceed in the following manner:

With, ie.

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