notes5-4 - SECTION 5.4 SOLUTIONS NEAR SINGULAR POINTS AND...

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SECTION 5.4 SOLUTIONS NEAR SINGULAR POINTS AND EULER EQUATIONS When P ( x ) ,Q ( x ) and R ( x ) are polynomials, then to find the singular points of P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0 , divide through by P ( x ), reduce the resulting fractions, and find the places where the reduced denominators are 0. These are generally nasty places for solutions of the differential equation. So why do we care about solutions near nasty places? OK, then, so we need to think about singular points. Are there differential equations with a singular point at x = 0 BUT which have nice solutions? 1. xy 0 - 5 y = 0 2. xy 0 - 19 y = 0 If this guess were to work for a second order differential equation L [ y ] = 0, what would L look like?
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An second order Euler equation has the form x 2 y 00 + αxy 0 + βy = 0, where α and β are constants. Notice that x = 0 is a regular singular point of such equations, so to begin with we look for solutions valid for x > 0. Guessing y = x r leads to an algebraic equation for r
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This note was uploaded on 02/14/2012 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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notes5-4 - SECTION 5.4 SOLUTIONS NEAR SINGULAR POINTS AND...

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