notes 5-4 - 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. Write down a simple-but-not-too- simple FIRST order linear de with a singular point at 0. Now solve it and ponder the solution. Do the same for a second order linear de.
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An 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 . EXAMPLE. Find the general solution of x 2 y 00 + 3 xy 0 - 15 y = 0 .
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