هارمونيك

May 25

هارمونيك

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and, taking antilogs (or raising each side to the power of e):

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There are several different ways of solving differential equations, which I'll list in approximate order of popularity. I'll also classify them in a manner that differs from that found in text books.

Know it or look it up. Of course! Very many differential equations have already been solved. Some of these you will learn, and others you can look up. This is by far the most common way by which scientists or mathematicians 'solve' differential equations. It is also how some (non-numerical) computer softwares solve differential equations.

Substitution. Often a differential equation can be simplified by a substitution for one or other of the variables. This may turn it into one that is already solved (see above) or that can be solved by one of the other methods. (The software packages do this, too.) This category of solution includes a range of techniques that you will learn in a second year mathematics course.

Guess and try. Another very common method of solving differential equations: guess what the solution might be, substitute it and, if it is not a solution, or not a complete solution, modify the guess until one has a complete solution. This is used often – more often than you would guess from reading books and papers, where the process usually appears to be rather elegant. In many cases you know something about the system studied, which gives you a clue. Experience helps, too, of course. However, we'll see below that the guessing is sometimes easy.

Modify a simpler solution. If you know a solution to an equation that is a simplified version of the one with which you are faced, then try modifying the solution to the simpler equation to make it into a solution of the more complicated one.

Transformation. Some differential equations become easier to solve when transformed mathematically. This is the main use of Laplace transformations.

Numerical solution. If all the above fail, then an algorithm, usually implemented on a computer, can solve it explicitly, calculating the derivatives as ratios. This is usually a method of last resort, for two reasons. First, it only gives you the solution for one particular set of boundary conditions and parameters, whereas all the above give you general solutions. Second, it has limited precision: numerical derviatives are inherently noisy.

Integration. This technique is elegant but is often difficult (or impossible). Sometimes one can multiply the equation by an integrating factor to make the integration possible.

Special types. This vague title is to include special techniques that work for particular types of equations. This, too, is for study in higher year mathematics courses.

Analog solution. Some differential equations are easily solved by analog computers. These are extremely fast and so suited to 'real time' control problems. Their disadvantages are limited precision and that analog computers are now rare.

Below we show two examples of solution of common equations. They are simple, because they have only constant coefficients, but they are the ones you will encounter in first year physics. These equations could be solved by several of the means above, but we shall illustrate only two techniques.

Because this is a simple equation, let's solve it by integration.

For this equation, it is possible to separate the variables, i.e. to rearrange the equation so that one side involves only x and the other only t. Here, we obtain

where C is a constant of integration. (More about the log function and constants of integration on Physclips calculus.) The constant(s) of integration are usually found from the boundary conditions: which in this case means from knowledge of x at some value of t. For this example, suppose we know that, at time t = 0, x = x0. Substitution gives

The difference between two logs is the log of the ratio, so

and, taking antilogs (or raising each side to the power of e):

Source: http://www.animations.physics.unsw.edu.au/jw/DifferentialEquations.htm


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