Differential Equations Midterm #2 Review

## Differential Equations Midterm #2 Review

Leftovers
• §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 nth degree and will have n roots
• For repeated roots, use ert, t·ert, t2·ert, t3·ert, etc. as needed

Chapter 6: The Laplace Transform
(I know £ isn't the right symbol but it's the best I could get in HTML!)
• §6.1: Definition
• £{f(t)} = ∫0e-st·f(t) dt
• Transforms of various common functions: 1, eat, tn, sin(at), cos(at), eatsin(bt), eatcos(bt), tneat

• §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
• uc(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 uc functions.
• Some important Laplace transforms:     £{uc},     £{uc·f(t – c)},     and £-1{F(s – c)}

• §6.4: Discontinuous Forcing Functions
• Write the forcing function in terms of uc, 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 dτ(t),
• where dτ(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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