What are A B in differential equations




With the differential equations, the applicability of mathematics in science and technology is particularly evident. For example, in the case of straight-line movement, the equation s '(t) = v (t) applies to the speed v (t) and the distance covered s (t), where s' denotes the derivative according to the time variable t. The function s is called a solution of the differential equation s' = v (t).

In general, one calls an equation of the form y(n) = f (x, y, y ', ..., y(n-1)) a ordinary differential equation of the nth order. Here f is a real-valued continuous function on a subset G of R.n + 1. A solution is a n times on a real interval I. differentiable function y, which is made for each x I. satisfies both of the following conditions: The point (x, y (x), y '(x), ..., y(n-1)(x)) lies in G, and it is y(n)(x) = f (x, y (x), y '(x), ..., y(n-1)(x)).

We look at a few Examples where we use the abbreviation DGL for "differential equation":

  1. The equation y '= y is a 1st order DGL. You can see immediately that y (x) = ex is a solution. In fact, y isc(x) = c ex for each constant c R. a solution and it is I.=R..
  2. The equation y '' = - y is a 2nd order DGL. A particular The solution is y (x) = cos (x), because it is y '(x) = - sin (x) and therefore y' '(x) = - sin'x = -cos (x) = - y (x ). A general solution is the function y (x) = c1cos (x) + c2sin (x).
  3. The equation y '= -x-1 y is a first order equation with general solution y (x) & nbsp = c x-1, it is y '(x) & nbsp = & nbsp-c x-2& nbsp = & nbsp- x-1 y (x). By specifying a Initial value condition the solution can be made unique, for example y (1) = 27. Then you only have one solution, namely y (x) = 27 x-1
  4. The equation y '= a (x) y with a continuous function a has solutions of the form y (x) = c eA (x), where A is some antiderivative of a and thus satisfies A '= a. This equation is called a homogeneous linear DGL 1st order.
  5. Solutions of inhomogeneous linear differential equation y '= a (x) y + b (x), where b (x) is continuous and not zero, is obtained according to an idea by Lagrange (1736-1813) Variation of the constants. One makes an approach y (x) = c (x) eA (x), where A is an antiderivative of a, and we get y '(x) = a (x) y (x) + c' (x) eA (x). Comparison with the given DGL then gives the solutions given in IV below.
Note that an antiderivative is always only determined up to an additive constant C. The following table shows some of the simpler ones 1st order DGL types listed with possible solution methods.

You have to be very careful here, for example g (y) not equal to 0 in II and III must apply. In a lecture on ordinary differential equations, criteria about the existence of solutions and their uniqueness when given Initial value condition derived.

In the general solution of an ordinary differential equation of order n, there are n free constants. If n = 1, then y '= f (x, y) at every point P of every solution curve y through the ODEc(x) a slope is given (namely the slope of the tangent to the curve at point P). One then speaks of one Direction field, and in simple cases the solution can be read from the direction field. At the top of this page is the directional field of a DGL of type y '= f (x-1y) painted on.

From n = 2, instead of initial value conditions, one can also use Boundary conditions to determine the constants. If y '' = f (x, y, y ') is a second order equation, then, for example, y (x0) = y0 and y '(x0) = y1 an initial value condition and by y (x0) = y0 and y (x1) = y1 a boundary value condition is given. A distinction is made here between Initial value problems and Boundary value problems.

Are also examined Systems of differential equations, For example, a homogeneous linear system of the first order can be written using the matrix calculus as y '=A.(x)y, in which A.(x) is a matrix, and treat it with methods of linear algebra. The theory of eigenvalues ​​and eigenvectors is also included here.

Ordinary differential equations are already considered in the lectures Analysis II or III. After that you can still have lectures on ordinary and partial differential equations connect. In partial differential equations, the solution functions depend on more than one variable. In the case of two variables x, y, it can be written, for example, as f (x, y, z (x, y), zx, e.g.y, e.g.xx, e.g.xy, e.g.yy, ...) = 0, where the indices mean partial derivatives.