Question

A body executes damped forced vibration given by equation dx²/dt²​+2kdx/dt​+b²x=e⁻ᵏᵗsinwt. Solve the equation for both the cases when ∘ w²≠b²−k² ∘ w²=b²−k².

Answer 1

The equation of a damped forced vibration can be solved using the method of undetermined coefficients. There are two cases to consider: when ω²≠b²−k² and when ω²=b²−k².

Step-by-step explanation:

The given equation represents the motion of a damped mass-spring system. To solve this equation, we can use the method of undetermined coefficients. First, we assume that the solution has the form x(t) = Asin(ωt) + Bcos(ωt), and substitute it into the equation. By comparing the coefficients of sin(ωt) and cos(ωt), we can solve for A and B. We have two cases to consider:

1. Case 1: When ω²≠b²−k²
2. In this case, the general solution will have the form x(t) = e^(-kt/2m)(C1sin(ω1t) + C2cos(ω1t)) + Dsin(ωt) + Ecos(ωt), where ω1 = √(b² − 4mk)/2m and ω = √(k/m - (b² − 4mk)/4m²).
3. Case 2: When ω²=b²−k²
4. In this case, the general solution will have the form x(t) = e^(-kt/2m)(C1 + C2t) + Dsin(ωt) + Ecos(ωt), where ω = √(k/m - (b² − 4mk)/4m²).