Math 415 Homework 4 3.1.24 Determine the values of α for which all solutions of y 00 + (3-α ) y0-2( 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) y0 + ( 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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