Math 267, Section E2, 2009f Hailiang Liu December 4, 2009 Chapter - - PowerPoint PPT Presentation

Math 267, Section E2, 2009f

Hailiang Liu December 4, 2009

Chapter 1: First Order Equations pp1-99 Chapter 2: Linear equations of higher order, pp100-193 Chapter 3: Power Series Methods pp194- Chapter 4: Laplace Transform methods pp266– 325 Chapter 5: Linear Systems of Differential Equations pp 326–429 Chapter 6: *Numerical Methods pp 430–479 Chapter 7: Nonlinear Systems and Phenomena pp480–554

Schedule reminder

◮ Review on your own on Dec. 7, 8, 10 ◮ You may ask questions these days: 10:00am–11:00am

• Dr. Jun PAN, Carver 384

◮ Last review session will be in classroom on Dec. 11;

12:10pm–1:00pm

◮ Final is to be held on Dec. 15, 12:00pm–2:00pm. ◮ Final involves total 8 problems, 100 points, 120 minutes.

Chapter 1: Main topics

dy dx = f (x, y),

• r

M(x, y)dx + N(x, y)dy = 0.

◮ Types of equations

Integrable equation Separable equation Linear equation Exact equation

◮ Methods to solve each type ... ◮ Substitution methods and equation reduction.

Integrable equations

◮ Mathematical form

dy dx = f (x) Its solutions is y =

• f (x)dx + C.

◮ Remember to add an integral constant. ◮ Review of how to integrate a given function.

Separable equations

◮ Mathematical form

dy dx = f (x)g(y) Its solutions is

• dy

g(y) =

• f (x)dx + C.

◮ Be sure no loss of solutions when separating variables

Linear equation

dy dx + P(x)y = Q(x)

◮ Integrating factor

ρ(x) = e

R P(x)dx ◮ Multiply the equation by ρ to obtain

d dx [ρy] = Q(x)ρ(x)

◮ Integration gives

y = ρ−1

• Q(x)ρ(x)dx + C
• .

Exact equation

M(x, y)dx + N(x, y)dy = 0.

◮ Definition: the equation is exact if there exists an F such that

M(x, y)dx + N(x, y)dy = d(F(x, y)), Its solutions is F(x, y) = C.

◮ Criterion: If My = Nx, then the equation must be exact. ◮ How to find F:

i) Regroup terms and use the product rule to observe F; ii) Integrate Fx = M to obtain F =

• Mdx + g(y)

Then N = Fy = ∂

∂y

• Mdx + dg

dy .

Substitution methods

◮ y′ = F(ax + by + c),

v = ax + by + c, then dv dx = a + bF(v).

◮ Homogeneous equation

y′ = F y x

• ,

v = y/x.

◮ The Bernoulli equation:

y′ + P(x)y = Q(x)yn, v = y1−n.

Reducible equations

◮ no explicit ’y’

F(x, y′, y′′) = 0 then p = y′ → F(x, p, p′) = 0.

◮ No explicit ’x’

F(y, y′, y′′) = 0 then p = y′ → F(y, p, p dp dy ) = 0.

Exam 1– 9/14/09

• 1. (5 points) Determine the order of the given differential

equations. (a) d2y

dt2 + sin(t + y) = sin t

(b) (dy

dt )2 + ty3 = 0

• 2. (5 points) Is the function y = 3t2 a solution of the differential

equation ty′ − y = t2? Verify your answer.

• 3. (10 points) Find the general solution of the equation

y′ = 4t − 6, and then find the particular solution with the initial condition y(0) = 1.

• 4. (15 points) Find the value of a for which the equation

(2xy2 + ay) + (2x2y + 2x)y′ = 0 is exact, and then solve it using that value of a.

• 5. (15 points) What type is the equation x2y′ + 2xy = 5y3?

Find its general solution.

Chapter 2: Main topics

Ly := y(n) + p1(x)y(n−1) + · · · + pn−1(x)y′ + pn(x)y

◮ nth order homogeneous linear equations Ly = 0 and solution

structures yc = c1y1 + · · · cnyn.

◮ Characteristic equation method y = erx for finding yi(x) ◮ Nonhomogeneous euqations Ly = f (x) and solution structures

y = yc(x) + yp(x).

◮ How to find yp for pi constants and f special.

Solution structures

◮ Superposition of solutions

L[ay1 + by2] = aL[y1] + bL[y2] = 0.

◮ Linear independence and Wronskians W = W (y1, · · · yn) ◮ Gneeral solution of homogeneous linear equations

yc = c1y2 + · · · cnyn.

◮ General solution of non-homogeneous solution

y = yc(x) + yp(x).

Find yc for Ly = 0

◮ Write down the characteristic equation

rn + p1rn−1 + · · · pn−1r + pn = 0.

◮ r is real, so y = erx is a solution ◮ r is complex, so two solutions are

y1 = Re[erx], y2 = Im[erx].

◮ If r repeated of multiplicity k, then

y1 = erx, y2 = xerx, · · · yk = xk−1erx.

Find yp by method of underdetermined coefficients

◮ The method is valid only if pi are constants and f is either a

polynomial in x; an exponential erx; coskx or sinkx, or linear combinations of them.

◮ Try solution of the same form

yp = xs˜ f (x). Here s is chosen so that no term in this solution appears in yc.

◮ Method of variation of parameters

yp = u1(x)y1 + u2(x)y2 and then try to find ui.

Exam 2– 10/02/09

• 1. [10 points.] Show that y1(x) = e−4x and y2(x) = xe−4x are two

linearly independent solutions for y ′′ + 8y ′ + 16y = 0, then find a solution satisfying y(0) = 2 and y ′(0) = −1.

• 2. [10 points.] Find the general solution of

y ′′′ − y ′′ + y ′ − y = 0.

• 3. [10 points.] Consider the following linear, non-homogeneous

equation, y ′′ + 4y = f (x). For each of the following cases, write and appropriate form of a particular solution (you do not need to find the exact solution.)

• a. f (x) = x2e2x
• b. f (x) = x cos(2x)
• c. f (x) = x − 1 + ex sin(2x)
• 4. [10 points.] Find the solution to the following initial value problem,

y ′′ − y = ex, y(0) = 3, y ′(0) = 15 2 .

Chapter 3: Main topics

◮ What is a power series, its convergence radius ◮ Series solution near ordinary points

y =

∞

• n=0

cn(x − a)n.

◮ The radius if convergence= at least as large as the distance

from a to the nearest singular point of the equation.

◮ Steps to find cn — two types of recurrence relations

Two-term recurrence More-term recurrence

Chapter 4: Main topics

◮ Laplace transform of typical functions and their inverse ◮ Properties of Laplace transform ◮ Transformation of Initial value problems ◮ Translation and partial fractions ◮ Derivatives, integrals, products of transforms ◮ Piecewise continuous ◮ Impulse and Delta functions

Laplace Transform

◮ F(s) = L{f (t)} =

∞

0 e−stf (t)dt,

s > a.

◮ Laplace of {1, eat, ta, tn, coskt, sinkt, u(t − a)} =? ◮ Any linear combinations can be transformed by using

L{af + bg} = aL{f } + bL{g}.

◮ Inverse Transforms

f (t) = L−1{F(s)}.

◮ Existence and Unique Theorem: If f is piecewise

continuous and |f | ≤ Mect, then F(s) exists for all s > c and is unique. Moreover, lim

s→∞ F(s) = 0.

Transform of IVPs

◮ Transform of derivatives

L{f ′(t)} = sF(s) − f (0), L{f ′′(t)} = s2F(s) − sf (0) − f ′(0)

◮ Transform of ax′′ + bx′ + cx = f (t) with x(0) = x0,

x′(0) = x1 is L{x} = F(s) + a(sx0 + x1) + bx0 as2 + bs + c .

◮ Transform of Integrals

L t f (τ)dτ

• = F(s)

s .

Translation and partial fractions

◮ Translation in s

L{eatf (t)} = F(s − a).

◮ Transform of eat{tn, coskt, sinkt} ◮ Decomposition of a fraction

P(s)/Q(s) = A1 s − a + A2 + B2s (s − a)2 + b2 + ...

◮ A practice example: solve

x′′ + 4x′ + 4x = t2; x(0) = x′(0) = 0.

Useful Transform formulas

◮ Convolution

L{f ∗ g} = L{f } · L{g}.

◮ Differentiation of Transforms

L{tnf (t)} = (−1)nF (n)(s).

◮ Integration of transforms

L f (t) t

• =

∞

s

F(τ)dτ.

◮ Three examples: find

(i)L−1

• 2

(s − 1)(s2 + 4)

• (ii)L−1{tan−1(1/s)}

(iii)L sinht t

Transform of special functions

◮ Piecewise continuous

L{u(t − a)f (t − a)} = e−asF(s).

◮ Periodic function* f (t + p) = f (t)

F(s) = 1 1 − e−ps p e−stf (t)dt.

◮ Pulse

L{δ(t − a)} = e−as.

◮ Practices:

L−1 e−as s3

• x′′ + 4x = δ(t − 2π); x(0) = 1, x′(0) = 1.

Exam 3– 10/22/09

• 1. [10 points.] Using power series methods to solve the equation

y′′ + xy = 0.

• 2. [10 points.] Show that the equation

xy′ + y = 0. has no non-trivial power series solution of the form y = ∞

n=0 cnxn.

• 3. [10 points.] Apply the definition to find the Laplace transform
• f the function ekt; then use the obtained result and the

relation coskt = eikt+e−ikt

2

to find L{coskt}.

• 4. [10 points.] Using Laplace transform to solve the initial value

problem x′′ − x = 5; x(0) = x′(0) = 0.

• 5. [10 points.] Using the formula L{tf (t)} = −F ′(s) to find

L−1 {ln(s − 1)} , s > 1.

Chapter 5: Main topics

◮ How to rewrite a higher equations into a first order system; ◮ The method of elimination (to reduce to higher order

equation to solve)

◮ Solution structure of linear systems ˙

x = P(t)x + f (t): x = c1x1 + · · · cnxn + xp(t).

◮ Eigenvalue method x = veλt ◮ Matrix exponentials eAt ◮ Nonhomogeneous system ˙

x = Ax + f (t).

Eigenvalue Method

◮ Single eigen-pairs (λi, vi), we have xi(t) = vieλit

|A − λiI| = 0, (A − λiI)vi = 0. Then the general solution is x = c1x1 + · · · + cnxn.

◮ For a complex eigen-pair (λ, v):

x1 = Re[veλt], x2 = Im[veλt]

◮ For repeated eigenvalue λ, find a generalized eigenvector v

x = eλt

• v + t(A − λI)v + t2

2 (A − λI)2v + ...

• ,

(A−λI)rv = 0.

Exponential matrix

◮ Let x1, · · · xn be n linearly independent solutions, then

Φ(t) = [x1, · · · , xn] is a fundamental matrix (Wronskian |Φ(t)| = 0). The general solution is x = Φ(t)c.

◮

eAt = Φ(t)Φ(0)−1.

◮ The solution with initial data x(0) = x0 is

x(t) = eAtx0.

Non-homogeneous linear system

◮ Variation of parameters: Let Φ(t) be a fundamental matrix of

x′ = P(t)x. Then a particular solution of x′ = P(t)x + f (t) is xp(t) = Φ(t)

• Φ(t)−1f (t)dt.

◮ If eAt is known for x′ = Ax + f (t) with x(0) = x0, then

x(t) = eAtx0 + t e−A(s−t)f (s)ds.

Exam 4– 11/12/09

1 (10 points) Transform the following system into an equivalent first order system, and then express it using matrix and vectors. x′′ = −6x + 2y y′′ = x − y + 10sint. 2 (10 points) Find a general solution of the system x′ = Ax with A = 4 −3 3 4

• .

3 (10 points) Find two linearly independent solutions of the system x′ = Ax with A = 1 −3 3 7

• .

4 (10 points) Find a particular solution of x′ = Ax + f (t) with A = −1 1

• ,

f = [sect, 0]⊤, eAt = cost −sint sint cost

• .

Chapter 7: Main topics

◮ Equilibrium solutions, critical points, asymptotic behavior ◮ Stability and phase plane ◮ Linear systems and almost system ◮ Applications*

One equation x′ = f (x)

◮ Critical points {x∗|

f (x∗) = 0}; x = x∗ is called an equilibrium solution,

◮ Stability: for each ǫ > 0, there exists δ > 0 such that

|x0 − x∗| < δ → |x(t) − x∗| < ǫ.

◮ If f ′(x∗) < 0, then x∗ is stable; else unstable. ◮ Phase diagram is simple.

2 × 2 Linear system: x′ = F(x, y), y ′ = G(x, y)

◮ Critical points

{(x∗, y∗)|F(x∗, y∗) = G(x∗, y∗) = 0}

◮ Phase portrait: draw a phase portrait for

x′ = x − y, y′ = 1 − x2.

◮ Critical point behavior: node (proper, improper), saddle,

center, spiral.

◮ Stability: sink or source ◮ Spiral toward a closed trajectory*

x′ = −ky + x(1 − x2 − y2), y′ = kx + y(1 − x2 − y2).

Linear system

◮ Linear system

x y ′ = A x y

• ,

A = a b c d

• .

◮ Critical points of linear system: eigenvalues of A:

node = real, same sign; saddle= real, opposite sign; spiral=complex, center=pure imaginary

◮ Stability:

(i) asymptotic stable, Re(λi) < 0; (ii) Stable Re(λi) = 0; (iii) unstable either of Re(λi) is positive.

Almost linear system

◮ Let (x∗, y∗) be a critical point for x′ = F, y′ = G, then

(u, v) = (x − x∗, y − y∗) solves

u v ′ = J u v

• +

r(u, v) s(u, v)

• ,

J = Fx(x∗, y ∗) Fy(x∗, y ∗) Gx(x∗, y ∗) Gy(x∗, y ∗)

• ◮ Same as linear system near critical points except two cases

(i) λ1 = λ2 (node or spiral) (ii) Pure imaginary (center or spiral).

◮ A practice example: Determine the type of the critical point

(4, 3) of the almost linear system x′ = 33 − 10x − 3y + x2; y′ = −18 + 6x + 2y − xy.

Exam 5*– 12/04/09

1 (10 points) Find critical point for x′ = x2 − 5x + 4, determine their stability and draw the phase diagram. 2 (10 points) Find the equilibrium solution of x′′ + 2x′ + 2x = 0 and determine its type and stability property. 3 (10 points) Classify the critical point as to type and stability x′ = x − 2y − 8, y′ = x + 4y + 10. 4 (10 points) Determine the type and stability of the critical point of the almost linear system x′ = 4x + 2y + 2x2 − 3y2, y′ = 4x − 3y + 7xy.