Transcription of 7.4 Cauchy-Euler Equation - University of Utah
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7.4 Cauchy-Euler Equation - University of Utah
Cauchy-Euler EquationThe differential equationanxny(n)+an 1xn 1y(n 1)+ +a0y= 0is called theCauchy-Eulerdifferential Equation of ordern. The sym-bolsai,i= 0, .. , nare constants andan6= Cauchy-Euler Equation is important in the theory of linear differ-ential equations because it has direct application toFourier s methodin the study of partial differential equations . In particular, the secondorder Cauchy-Euler equationax2y +bxy +cy= 0accounts for almost all such applications in applied second argument for studying the Cauchy-Euler Equation is theoret-ical: it is a single example of a differential Equation with non-constantcoefficients that has a known closed-form solution.
Back-substitute x = et and t = lnjxjin this equation to obtain two independent solutions of 2x2y00+ 4xy0+ 3y = 0: x 1=2 cos p 5 2 lnjxj!; e t=2 sin p 5 2 lnjxj!: Substitution Details. Because x = et, the factor e t=2 is written as (et) 1=2 = x 1=2. Because t = lnjxj, the trigonometric factors are back-substituted like this: cos p 5 2 t = cos p ...
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