Question: SHOW ALL WORK AND EXPLAIN STEPS: Suppose is a constant and consider the differential equation: y''+ y' + y = 0 (a) Determine the characteristic

SHOW ALL WORK AND EXPLAIN STEPS:

Suppose is a constant and consider the differential equation: y''+ y' + y = 0

(a) Determine the characteristic (auxiliary) equation for this differential equation and its roots.

(b)

For what values of will there be two distinct real roots to the characteristic equation?

What is the general solution to the differential equation for these values of ?

(c)

For what values of will there be one repeated real root to the characteristic equation?

What is the general solution to the differential equation for these values of ?

(d)

For what values of will there be complex roots to the characteristic equation?

What is the general solution to the differential equation for these values of ?

Note: For parts (b) - (d), your answers will involve the parameter .

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