Problem 2: For an underdamped system (3< wo), the two solutions can be written asx₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin wit. (a) Show that as → wo, i(t) approachesthe critically damped solution e-Bt. (b) What happens to r₂(t)? Show that the expressionx₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the secondcritically damped solution te-ßt in the limit ß→wo.

Answered: Image /qna-images/answer/8f88dac9-2abc-403a-bf5b-09124eaaac67.jpg

Problem 2: For an underdamped system (8< wo), the two solutions can be written as x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin w₁t. (a) Show that as →→ wo, x₁(t) approaches the critically damped solution e-ßt. (b) What happens to r₂(t)? Show that the expression x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second critically damped solution te-ßt in the limit 3 → wo.

Problem 2: For an underdamped system (8< wo), the two solutions can be written as x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin w₁t. (a) Show that as →→ wo, x₁(t) approaches the critically damped solution e-ßt. (b) What happens to r₂(t)? Show that the expression x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second critically damped solution te-ßt in the limit 3 → wo.