Differential Equations Test #2 Review

• General case: a·y″ + b·y′ + c·y = g(t)
  • Homogeneous if g(t)=0; nonhomogeneous otherwise; g(t) is the nonhomogeneity.
  • General solution will involve two unknown constants c₁ & c₂.
  • For specific solution, need initial values for y(t₀) & y′(t₀); plug in and solve for c₁ & c₂.
  • Individual solutions are linearly independent if their Wronskian is nonzero.
    • Wronskian Determinant: W(y₁, y₂) = y₁y₂′ − y₂y₁′
• Solving Homogeneous Equations: a·y″ + b·y′ + c·y = 0
  • Any linear combination of solutions is also a solution.
  • General solution can be written y = c₁y₁ + c₂y₂, where y₁ & y₂ are individual solutions.
  • Set up the characteristic quadratic a·r² + b·r + c = 0 and solve for r.
    • §4.2: If two real roots r₁ & r₂ exist, y₁ = e^r₁t and y₂ = e^r₂t.
    • §4.2: If one repeated root r exists, y₁ = e^rt and y₂ = t·e^rt.
    • §4.3: If two complex roots α±βi exist, y₁ = e^αtcos(βt) and y₂ = e^αtsin(βt).
      • The reason for this is Euler's Formula: e^A+Bi = e^A·e^Bi = e^A·(cosB + i·sinB).
• Solving Nonhomogeneous Equations: a·y″ + b·y′ + c·y = g(t) ≠ 0
  • §4.5: The nonhomogeneous general solution can be written as y = c₁y₁ + c₂y₂ + Y_P,
    • where c₁y₁ + c₂y₂ is the solution of the corresponding homogeneous equation
    • and Y_P is any single "particular solution" of the nonhomogeneous equation.
    • The difficulty of course lies in finding Y_P, since we already can find y₁ and y₂.
  • §§4.4-5: The Method of Undetermined Coefficients
    • This method is useful if g(t) is a sine, cosine, polynomial, or exponential function.
      • (or a sum or product of these types of functions)
    • Assume that Y_P has a similar form to g(t), with an unknown coefficient A.
      • For example, if g(t) = e^3t, use Y_P = Ae^3t.
      • If g(t) is a sine or cosine, use Y_P = Asin + Bcos to cover all possibilities.
      • If g(t) is a solution of the corresponding homogeneous equation, try t·g(t).
    • Substitute Y_P and its derivatives into original equation and solve for A, B, C, etc.
    • Add Y_P to solution of corresponding homogeneous equation for general solution.
    • Principle of Superposition: if g(t) is a sum, find Y_P for each term and add them.
  • §4.6: The Method of Variation of Parameters
    • This method is more general, but less practical (often involves messy integrals).
    • Find homogeneous solution c₁y₁ + c₂y₂; assume Y_P = v₁y₁ + v₂y₂.
      • v₁ and v₂ represent unknown functions instead of constants.
    • Find Y_P′; assume for simplicity that y₁v₁′ + y₂v₂′ = 0; find Y_P″.
    • Substitute Y_P, Y_P′, Y_P″ into original equation and simplify.
    • This and y₁v₁′ + y₂v₂′ = 0 form a system of equations; solve for v₁′ & v₂′.
    • Integrate to find v₁ & v₂, then substitute into Y_P and find solution.

• Higher order homogeneous linear equations can be solved in the same way!
  • §6.2: Set up and solve a characteristic polynomial: a_nrⁿ + a_n−1rⁿ⁻¹ + ... + a₁r + a₀ = 0.
    • The Rational Roots Theorem and synthetic division can be very useful here.
    • Individual solutions will, as before, be of the form e^rt.
    • For repeated roots of higher multiplicity, solutions are e^rt, te^rt, t²e^rt, etc.
    • Complex roots still lead to e^αtcos(βt) and e^αtsin(βt).
      • (possibly times t, t², etc. for repeated complex roots)
  • §6.1: Some interesting theoretical mumbo-jumbo for the general case...
    • For an n^th order equation, there should be a fundamental set of n solutions.
    • This also means n unknown coefficients c₁,...,c_n, so n initial conditions are needed!
    • The Wronskian W(y₁,...,y_n) can still be used to determine if the solutions
      • are linearly independent, but it will be an nxn determinant.

• §3.5: Circuits with resistors, capacitors, and inductors
  • Kirchoff's Voltage Law: total voltage provided = total voltage used (for a closed loop).
  • Voltage provided: E(t), which might be constant (DC), variable (AC), or zero (nothing).
  • Voltage used: total sum of voltage drop across each circuit element, including:
    • Resistors: E_R = I·R, where R is resistance (constant)
    • Capacitors: E_C = q/C, where C is capacitance (constant)
    • Inductors: E_L = L·dI/dt, where L is inductance (constant)
  • Substituting these into Kirchoff's Law yields a differential equation!
  • Important note: I (current) is the flow rate of charge (q), so I = q′ and dI/dt = q″.
  • Initial conditions may include charge, current, or rate of change in current.
• §4.1, §4.9: Mechanical Vibration (mass on a spring)
  • Important constants: m = mass, b = damping coefficient, k = spring constant.
  • Generally: m·y″ + b·y′ + k·y = F_ext(t)
    • This is homogeneous if there is no external force!
  • Three possibilities, depending on the discriminant (b²−4mk):
    • b²−4mk > 0: overdamped (two real roots; decaying solution)
    • b²−4mk = 0: critically damped (repeated root; partial oscillation then decay)
    • b²−4mk < 0: underdamped (complex roots; oscillations with gradual decay)
  • Initial conditions will be the object's position and velocity at time t=0.

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