- §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}, t^{2}·e^{rt}, t^{3}·e^{rt}, etc. as needed

- Everything works the same as second order, but more complicated.

(I know £ isn't the right symbol but it's the best I could get in HTML!)

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

- £{f(t)} = ∫
- §6.2: Initial Value Problems
- £{y'} = s·£{y} – y(0), £{y"} = s
^{2}·£{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.

- £{y'} = s·£{y} – y(0), £{y"} = s
- §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)}

- u
- §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.

- Write the forcing function in terms of u
- §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 – t
_{0}) is "infinite" at t = t_{0}, zero everywhere else, and its integral is 1. - Laplace transform: £{δ(t – t
_{0})} = e^{-st0} - This is mainly useful for approximating a short sharp shock at time t
_{0}whose total impulse is 1.

- The Dirac delta function δ(t) is defined as the limit as τ → 0 of d