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notes5-4

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

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