Final answer:
To determine the values of a for which all solutions tend to zero or become unbounded as t->infinity, we can use the characteristic equation and analyze the discriminant.
Step-by-step explanation:
To determine the values of a for which all solutions of the differential equation y''-(2a-12)y'+(a^2-12a+20)y = 0 tend to zero as t->infinity, we can use the characteristic equation. For a solution to tend to zero as t->infinity, the characteristic equation must have only negative real roots or complex conjugate roots with negative real parts. In this case, the characteristic equation is:
r^2-(2a-12)r+(a^2-12a+20) = 0
We can solve this quadratic equation using the quadratic formula:
r = (-(2a-12) ± √((2a-12)^2-4(a^2-12a+20)))/(2)
Analyze the discriminant to determine the values of a. If the discriminant is negative, there will be complex conjugate roots with negative real parts and all solutions will tend to zero as t->infinity. If the discriminant is zero or positive, there will be real roots and we need to check further conditions to determine the behavior of the solutions as t->infinity.
To determine the values of a for which all (nonzero) solutions become unbounded as t->infinity, we can examine the discriminant as well. If the discriminant is positive, there will be real roots with at least one positive real root and the solutions will become unbounded as t->infinity. If the discriminant is zero or negative, all (nonzero) solutions will not become unbounded as t->infinity.