University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

265

Pages:

6
Author:

Jabin F.

2nd Order differential equation with constant coefficient: a₂d²y/dx² + a₁dy/dx + a₀y = f(t) where a₂, a₁, a₀ are constants Solution: y = C.F. + P.I. → Complementary function → Particular Integral C.F. is obtained from roots of equation: D² + a₁D + a₀ = 0 where D = d/dx and λ₁, λ₂ are roots If roots are real and different: C.F. = c₁e^(λ₁x) + c₂e^(λ₂x) If roots are real and equal: C.F. = (c₁ + c₂x)e^(λx) Note: For cubic equation, 3 roots are obtained λ₁, λ₂, λ₃ C.F. = c₁e^(λ₁x) + c₂e^(λ₂x) + c₃e^(λ₃x) P.I. P.I. is needed if f(t) ≠ 0 P.I. = 1/f(D) * f(t) where f(D) = D² + a₁D + a₀ f(D) is an operator If f(t) = e^(αx) then P.I. = e^(αx) / f(α) when f(α) ≠ 0 If f(α) = 0 then P.I. = xe^(αx) / f'(α) when f'(α) ≠ 0 and so on... If f(x) = Sin(ax) or Cos(ax) then... P.I. = 1/f(D) * Sin(ax) If f(α) = 0 then P.I. = x Sin(ax) / f'(α) when f'(α) ≠ 0 Example: Solve: d²x/dt² + w²x = 0 Solution: Auxiliary eq: D² + w² = 0 ⇒ D = ±iw Solution: x = Ae^(iwt) + Be^(-iwt) using (e^(iθ) = Cosθ + iSinθ) Euler's formula x = A[Cos(wt) + iSin(wt)] + B[Cos(wt) - iSin(wt)] x = (A+B)Cos(wt) + i(A-B)Sin(wt) Let new constants: A+B = C₁ i(A-B) = C₂ x = C₁Cos(wt) + C₂Sin(wt) if we substitute C₁ = x₀Cos(φ) & C₂ = x₀Sin(φ) x = x₀[Sin(φ)Cos(wt) + Cos(φ)Sin(wt)] x = x₀Sin(φ+wt) where x₀ & φ are unknown constants.