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Unformatted text preview: 19 Euler Equations We now know how to completely solve any equation of the form ay + by + cy = or even a y ( N ) + a 1 y ( N 1 ) + + a N 2 y + a N 1 y + a N y = in which the coefficients are all constants (provided we can completely factor the corresponding characteristic polynomial). Let us now consider some equations of the form ay + by + cy = or even a y ( N ) + a 1 y ( N 1 ) + + a N 2 y + a N 1 y + a N y = when the coefficients are not all constants. In particular, let us consider the Euler equations, described more completely in the next section, in which the coefficients happen to be particularly simple polynomials. 1 As with the constant-coefficient equations, we will discuss the second-order Euler equations (and their solutions) first, and then note how those ideas extend to corresponding higher order Euler equations. 19.1 Second-Order Euler Equations Basics A second-order differential equation is called an Euler equation if it can be written as x 2 y + xy + y = where , and are constants (in fact, we will assume they are real-valued constants). For example, x 2 y 6 xy + 10 y = , 1 These differential equations are also called Cauchy-Euler equations 395 396 Euler Equations x 2 y 9 xy + 25 y = , and x 2 y 3 xy + 20 y = are the Euler equations well solve below. In these equations, the coefficients are not constants but are constants times the variable raised to the power equaling the order of the corresponding derivative. Notice, too, that the first coefficient, x 2 , vanishes at x = 0 . This means we should not attempt to solve these equations over intervals containing 0 . For convenience, we will use ( , ) as the interval of interest. You can easily verify that the same formulas derived using this interval also work using the interval ( , ) after replacing the x in these formulas with either x or | x | . Euler equations are important for two or three good reasons: 1. They are easily solved. 2. They occasionally arise in applications, though not nearly as often as equations with constant coefficients. 3. They are especially simple cases of a broad class of differential equations for which infinite series solutions can be obtained using the method of Frobenius. 2 (Whether or not this is a good reason may depend on your point of view.) The basic approach to solving Euler equations is similar to the approach used to solve constant-coefficient equations: assume a particular form for the solution with one constant to be determined, plug that form into the differential equation, simplify and solve the resulting equation for the constant, and then construct the general solution using the constants found and the basic theory already developed....
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