Differential Equations problem help needed

Question

Classify the values of β for which the characteristic equation has: distinct real roots, repeated real roots, and complex roots.

y"-4y'+(4-β)y=0

y(0)=1, y'(0)=1

Answer 1

Write the characteristic equation: r² - 4r +(4-β) = 0 (subbing in y = e^rx)

This has two distinct roots when the discriminant sqrt((-4)² -4(4-β)) is nonzero and real. This is equivalent to

4β ≥ 0

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