H4_130 - Math 415 Homework 4 3.1.24 Determine the values of...

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Math 415 Homework 4 3.1.24 Determine the values of α for which all solutions of y 00 + (3 - α ) y 0 - 2( α - 1) y = 0 tend to zero as t → ∞ ; also determine the values of α for which all (nonzero) solutions become unbounded as t → ∞ . The characteristic equation for the given differential equation is 0 = r 2 +(3 - α ) r - 2( α - 1) = ( r +2)( r - ( α - 1)). So the general solution is y ( t ) = C 1 e - 2 t + C 2 e ( α - 1) t . The e - 2 t term goes to zero as t approaches infinity, so we only need to concern ourselves with the second term. If α - 1 < 0 (i.e. α < 1), then the solution will approach zero as t → ∞ . If α > 1 then the solution will become unbounded as t → ∞ (whether to positive or negative infinity depends on C 2 ). 3.2.25 Verify that y 1 ( x ) = x and y 2 ( x ) = xe x are solutions of the differential equation x 2 y 00 - x ( x + 2) y 0 + ( x + 2) y = 0 (with x > 0). Do they constitute a fundamental set of solutions? For y 1 we get: x 2 · 0 - x ( x + 2) · 1 + ( x + 2) x = 0 . And for y 2 we get: x 2 ( xe x + 2 e x ) - x ( x + 2)( xe x + e x ) + ( x + 2) xe x = ( x 3 + 2 x 2 - x 3 - x 2 - 2 x 2 - 2 x + x 2 + 2 x ) e x = 0 . So both are solutions to the differential equation. To see that they form a fundamental set of solutions, we check the Wronskian of the two:
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This note was uploaded on 07/26/2011 for the course MATH 415 taught by Professor Costin during the Spring '07 term at Ohio State.

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