notes3 - z 2 = e r 2 t ) y 1 = x r 1 and y 2 = x r 2 . If r...

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MATH 1005A - Notes 3 Cauchy-Euler Equations An equation of the form x 2 y 00 + Axy 0 + By =0 ; where A and B are constants (or, equivalently, ax 2 y 00 + bxy 0 + cy =0 ,w i th a; b and c constants), is called a Cauchy-Euler equation. In standard form, the equation is y 00 + A x y 0 + B x 2 y =0 : Since p ( x )= A x and q ( x )= B x 2 are unde¯ned at x = 0, the solution may be unde¯ned at x = 0. Thus, we assume that x 6 = 0. A Cauchy-Euler equation can be transformed into a constant-coe±cient equation as follows: For x> 0, let x = e t and y ( x )= z ( t ). Then t =ln( x ) and, by the chain rule, dy dx = dz dt dt dx = 1 x dz dt ; d 2 y dx 2 = ¡ 1 x 2 dz dt + 1 x d 2 z dt 2 dt dx = ¡ 1 x 2 dz dt + 1 x 2 d 2 z dt 2 ; and the equation x 2 y 00 + Axy 0 + By =0becomes d 2 z dt 2 ¡ dz dt ¸ + A dz dt + Bz =0 ,or z 00 +( A ¡ 1) z 0 + Bz =0 ; which has constant coe±cients. If z 1 ( t )and z 2 ( t ) are two independent solutions of z 00 +( A ¡ 1) z 0 + Bz =0 ,th entw o independent solutions of x 2 y 00 + Axy 0 + By =0areg ivenby y 1 ( x )= z 1 (ln x )a n d y 2 ( x )= z 2 (ln x ) : Since solutions of a constant-coe±cient equation are sought in the form z = e rt and y ( x )= z ( t )w ith t =ln( x ) ;y ( x )= e rt = e r ln( x ) = e ln( x r ) = x r . Thus, solutions of an Euler equation can be sought directly in the form y = x r . If r 1 6 =
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Unformatted text preview: z 2 = e r 2 t ) y 1 = x r 1 and y 2 = x r 2 . If r 1 = r 2 (real), then z 1 = e r 1 t and z 2 = te r 1 t ) y 1 = x r 1 and y 2 = x r 1 ln( x ). If r 1 ; r 2 = i (complex), then z 1 = e t cos( t ) and z 2 = e t sin( t ) ) y 1 = x cos[ ln( x )] and y 2 = x sin[ ln( x )]. For x < 0, let x = e t and y ( x ) = z ( t ). Then t = ln( x ), and the same equation for z ( t ) results. In either case, t = ln j x j . Since j x j = x; x > x; x < , replacing x by j x j gives the solutions for any x 6 = 0. Thus, 2 If r 1 6 = r 2 are real, then y 1 = j x j r 1 and y 2 = j x j r 2 . If r 1 = r 2 (real), then y 1 = j x j r 1 and y 2 = x r 1 ln j x j . If r 1 ; r 2 = i (complex), then y 1 = j x j cos( ln j x j ) and y 2 = j x j sin( ln j x j )....
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notes3 - z 2 = e r 2 t ) y 1 = x r 1 and y 2 = x r 2 . If r...

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