Differential Equations Midterm #2 Review

• §3.7: Harmonic Motion
  • m·u" + γ·u' + k·u = F(t), where m = mass, γ = damping, k = spring constant.
  • F(t) is the sum of any external forces (forcing function); zero if there are no other forces.
  • If γ=0 (undamped motion), solutions will be sinusoidal. Otherwise, sinusoidal with exponential decay.
  • Overdamped: damping too strong (discriminant < 0). Critically damped: discriminant = 0.


• §4.1 & §4.2: Higher Order Differential Equations
  • Everything works the same as second order, but more complicated.
    • Wronskian is an nxn determinant
    • Specific solution requires n points of initial conditions
    • For constant coefficients, characteristic polynomial is n^th degree and will have n roots
    • For repeated roots, use e^rt, t·e^rt, t²·e^rt, t³·e^rt, etc. as needed

• §6.1: Definition
  • £{f(t)} = ∫₀^∞e^-st·f(t) dt
  • Transforms of various common functions: 1, e^at, tⁿ, sin(at), cos(at), e^atsin(bt), e^atcos(bt), tⁿe^at


• §6.2: Initial Value Problems
  • £{y'} = s·£{y} – y(0),     £{y"} = s²·£{y} – s·y(0) – y'(0),     etc.
  • Many initial value problems can be solved more easily by taking the Laplace of both sides.
  • Simplify, solve for £{y}, and then invert the transform.


• §6.3: Step Functions
  • u_c(t) is the unit step function; it equals 0 for t∈[0,c) and 1 for t ≥ c.
  • Any piecewise function can be written in a single line through use of u_c functions.
  • Some important Laplace transforms:     £{u_c},     £{u_c·f(t – c)},     and £^-1{F(s – c)}


• §6.4: Discontinuous Forcing Functions
  • Write the forcing function in terms of u_c, and take the Laplace of both sides.
  • Decomposition into partial fractions is often useful.


• §6.5: Sudden Impulse Forcing Functions
  • The Dirac delta function δ(t) is defined as the limit as τ → 0 of d_τ(t),
  • where d_τ(t) = 1/(2τ) if t∈[-τ,τ], and zero everywhere else.
  • Thus δ(t – t₀) is "infinite" at t = t₀, zero everywhere else, and its integral is 1.
  • Laplace transform: £{δ(t – t₀)} = e^-st₀
  • This is mainly useful for approximating a short sharp shock at time t₀ whose total impulse is 1.

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