Review Outline for first quiz, Feb 24th

Alec Johnson

Methods of Solution of ODEs by classification

1. Nonlinear

(a) 1st order: y′ = f(t, y)

i. (§2.2) separation of variables ii. approximation methods

A. (§1.1) direction fields B. (§2.7) Euler’s method C. (§2.8) successive approximations

(b) 2nd order: y′′ = f(t, y, y′)

(c) nth order: y^{(n)} = f(t, y, y′, …, y^{(n−1)}) • (§7.1) convert to a first-order system and solve

2. Linear

(a) 1st order: y′ + p(t)y = g(t) i. integrating factors ii. short-cuts

A. homogeneous (g = 0): separate B. constant coefficients: y′ + by = g(t)

• undetermined coefficients

• Laplace transform

(b) 2nd order: L[h] = y′′ + p(t)y′ + q(t)y = g(t)

i. homogeneous: g=0

A. (§3.2) general solution is a linear combination c_{1}y_{1}(t) + c_{2}y_{2}(t), where y_{1}(t) and y_{2}(t) are linearly independent solutions.

• (§3.3) Wronskian determininant – definition of Wronskian

– functions are linearly independent if Wronskian is nonzero.

– solutions to ODE are linearly dependent if Wronskian is zero. B. constant coefficients: y′′ + by′ + cy = 0

guess e^{rt} get r^{2} + br + c = 0 factor: (r − r_{1})(r − r_{2}) = 0 Three cases:

• (§3.1) distinct real roots: r_{1}, r_{2}

• (§3.5) repeated (real) root: r_{1} = r_{2} Two independent solutions: y_{1}(t) = e^{rt} y_{2}(t) = te^{rt} • (§3.4) complex conjugate pair: r = λ ± iμ

Two independent real solutions: y_{1}(t) = e^{λt} cos(μt) y_{2}(t) = e^{λt} sin(μt)

C. nonconstant coefficients • (§3.5) reduction of order to get a second solution: Guess y(t) = u(t)y_{1}(t).

• (Chapter 5, not covered in this course) series solutions: plug a power series y =

∑_{k=1}^{∞} a_{n}t^{n} into the differential equation and solve the resulting difference equation.

ii. nonhomogeneous (g ≠ 0) Need a particular solution: L[Y]=g. A. constant coefficients

• (§3.6) undetermined coefficients Works for nice forcing functions. Forcing function: g(t) = P_{n}(t)e^{at}(a cos(βt) + b sin(βt) (P_{n} an n-th order polynomial). Guess: t^{s}[(A_{0}t^{n} + A_{1}t^{n−1} + ⋯ + A_{n}) cos(βt) + (B_{0}t^{n} + B_{1}t^{n−1} + ⋯ + B_{n}) sin(βt)]e^{at} where s=number of times α + iβ is a root of the characteristic equation.

• (Chapter 6, not covered on first quiz) Laplace transform Works for general forcing functions, including step functions and Dirac delta functions (spike functions).

B. nonconstant coefficients

• (§3.7) variation of parameters Given: y_{1} and y_{2} are solutions to L[y] = 0. Seek solution Y to L[Y] = g of the form Y = u_{1}(t)y_{1} + u_{2}(t)y_{2} subject to the requirement: u′_{1}y_{1} + u′_{2}y_{2} = 0 u′_{1}y′_{1} + u′_{2}y′_{2} = g Get: u_{1} = − ∫ \frac{gy_{2}}{W} u_{2} = ∫ \frac{gy_{1}}{W} where W = y_{1}y′_{2} − y′_{1}y_{2}.

(c) (Chapter 4) nth order (just like 2nd order)

i. homogeneous

A. (§4.2) constant coefficients Guess e^{rt}. • factoring characteristic polynomial over complex plane.

• Use polar form to find roots.

• Use complex conjugate roots to get real solutions. ii. nonhomogeneous

A. (§4.3) method of undetermined coefficients …