Agenda

Review

Reminders

Final ExamThursday, 12/148-10 am, CHEM 1800

WebHW 10 due 12/12

Office hours Tues, Thurs1-2 pm (5852 East Hall)

MathLab office hourSun 7-8 pm (MathLab)

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Find non-negative real numbers A, ω, and φ such that

Re

(ie2it

1 + i

)= A cos (ωt − φ).

Answer

A = 1/√

2, ω = 2, φ = 7π/4

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Find non-negative real numbers A, ω, and φ such that

Re

(ie2it

1 + i

)= A cos (ωt − φ).

Answer

A = 1/√

2, ω = 2, φ = 7π/4

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Let

A =

[a −22 1

],

and consider the homogeneous linear system x′ = Ax. For each of thefollowing conditions, determine all values of a (if any) which are suchthat the system satisfies the condition.

(a) Saddle

(b) Proper node (star)

(c) Stable node

(d) Asymptotically stable spiral. What is the direction of rotation?

(e) Unstable spiral

(f) Unstable defective node

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Let

A =

[a −22 1

],

and consider the homogeneous linear system x′ = Ax. For each of thefollowing conditions, determine all values of a (if any) which are suchthat the system satisfies the condition.

(a) Saddle a < −4

(b) Proper node (star) None

(c) Stable node −4 < a < −3

(d) Asymptotically stable spiral. What is the direction of rotation?−3 < a < −1, counterclockwise

(e) Unstable spiral −1 < a < 5

(f) Unstable defective node a = 5

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Parts (a)-(c) deal with the nonlinear autonomous system

x ′ = x2 − y2

y ′ = x2 + y2 − 8.

(a) Find the critical points for this system.

(b) There is one equilibrium point in the south-west quadrant. Find the Jacobian atthis equilibrium point.

(c) The equilibrium point you found in (b) is a stable spiral. For large t, the solutionswhich converge to this equilibrium have x-coordinates which arewell-approximated by the function Aeat cos (ωt − φ) for some constants A, φ, aand ω. Some of these constants depend upon the particular solution, and someare common to all solutions of this type. Find the values of the ones which arecommon to all such solutions.

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Parts (a)-(c) deal with the nonlinear autonomous system

x ′ = x2 − y2

y ′ = x2 + y2 − 8.

(a) Find the critical points for this system. (−2,−2), (−2, 2), (2,−2), (2, 2)

(b) There is one equilibrium point in the south-west quadrant. Find the Jacobian atthis equilibrium point.

J(−2,−2) =

[−4 4−4 −4

](c) The equilibrium point you found in (b) is a stable spiral. For large t, the solutions

which converge to this equilibrium have x-coordinates which arewell-approximated by the function Aeat cos (ωt − φ) for some constants A, φ, aand ω. Some of these constants depend upon the particular solution, and someare common to all solutions of this type. Find the values of the ones which arecommon to all such solutions.

a = −4, ω = 4

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

In (a)-(c) we consider the autonomous equation

x ′ = x3 − x2 − 2x .

(a) Sketch the phase line for this equation.

(b) Sketch the graphs of some solutions. Be sure to include atleast one solution with values in each interval above, below,and between the critical points.

(c) Suppose x(0) is quite small, say 0.1. For t > 0, x(t) is bestapproximated by 0.1eat for what value of a?

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

In (a)-(c) we consider the autonomous equation

x ′ = x3 − x2 − 2x .

(a) Sketch the phase line for this equation.

(b) Sketch the graphs of some solutions. Be sure to include at least one solution withvalues in each interval above, below, and between the critical points.

(c) Suppose x(0) is quite small, say 0.1. For t > 0, x(t) is best approximated by0.1eat for what value of a? a = −2

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

At what frequency ω = ωr does the steady state solution to

x ′′ + 2x ′ + 10x = cos (ωt)

have the largest amplitude?

Answer

ωr = 2√

2

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

At what frequency ω = ωr does the steady state solution to

x ′′ + 2x ′ + 10x = cos (ωt)

have the largest amplitude?

Answer

ωr = 2√

2

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Consider the IVP

y ′′ + y = fk(t), y(0) = 0, y ′(0) = 0,

where fk(t) = [u3−k(t)− u3+k(t)] /2k with 0 < k ≤ 1.

(a) Find the solution y = φ(t, k) of the IVP.

(b) Calculate limk→0 φ(t, k) from the solution found in part (a).

(c) Observe that limk→0 fk(t) = δ(t − 3). Find the solution φ0(t)of the given IVP with fk(t) replaced by δ(t − 3). Is it true thatφ0(t) = limk→0 φ(t, k)?

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Consider the IVPy′′ + y = fk (t), y(0) = 0, y′(0) = 0,

where fk (t) =[u3−k (t)− u3+k (t)

]/2k with 0 < k ≤ 1.

(a) Find the solution y = φ(t, k) of the IVP.

φ(t, k) =1− cos (t − 3 + k)

2ku3−k (t)−

1− cos (t − 3− k)

2ku3+k (t)

(b) Calculate limk→0 φ(t, k) from the solution found in part (a).Use

cos (t − 3 + k) = cos (t − 3) cos k − sin (t − 3) sin k

andcos (t − 3− k) = cos (t − 3) cos k + sin (t − 3) sin k

and carefully take the limit to show that

limk→0

φ(t, k) = sin (t − 3)u3(t).

(c) Observe that limk→0 fk (t) = δ(t − 3). Find the solution φ0(t) of the given IVP with fk (t) replaced byδ(t − 3). Is it true that φ0(t) = limk→0 φ(t, k)?

φ0(t) = sin (t − 3)u3(t)

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Suppose thatL = D3 + D.

(a) What is the transfer function of the operator L?

(b) What is the impulse response of this operator?

(c) What is the Laplace transform X (s) of the solution toL[x ] = cos (2t) with x(0) = x ′(0) = x ′′(0) = 0 as the initialconditions?

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Suppose thatL = D3 + D.

(a) What is the transfer function of the operator L?

H(s) =1

s3 + s

(b) What is the impulse response of this operator?

h(t) = 1− cos t

(c) What is the Laplace transform X (s) of the solution to L[x] = cos (2t) withx(0) = x ′(0) = x ′′(0) = 0 as the initial conditions?

X (s) = H(s)G(s) =1

s3 + s·

s

s2 + 4

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Suppose that a 3× 3 system of DE’s has the form

x′ = Ax

and that the eigenvalues of A are λ = 2± 3i and λ = −7.Describe the behavior of the phase portrait trajectories.

AnswerThere will exist a plane, that contains the origin, that has theproperty that any trajectory that starts in the plane will remainin the plane while spiraling away from the origin. Anytrajectory that starts outside of the plane will asymptoticallyapproach the plane while simultaneously spiraling away fromthe origin.

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Suppose that a 3× 3 system of DE’s has the form

x′ = Ax

and that the eigenvalues of A are λ = 2± 3i and λ = −7.Describe the behavior of the phase portrait trajectories.

AnswerThere will exist a plane, that contains the origin, that has theproperty that any trajectory that starts in the plane will remainin the plane while spiraling away from the origin. Anytrajectory that starts outside of the plane will asymptoticallyapproach the plane while simultaneously spiraling away fromthe origin.

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Indicate if each statement is true or false.

(a) Let f and g be differentiable for every x . If W [f , g ] = 0 for some x , then f and gmust be linearly dependent.

(b) Suppose we’d like to find a particular solution to y ′′ + y ′ + y = x2 sin x .According to the method of undetermined coefficients we should look for asolution of the form yp = (Ax2 + Bx + C)(D cos x + E sin x).

(c) Suppose that x1(t) and x2(t) are solutions of x′ = Ax. It is possible forW [x1, x2] = t2 − 4?

(d) A real 5× 5 matrix cannot have 5 complex eigenvalues.

(e) Variation of parameters is the only method we’ve learned that can find aparticular solution to y ′′ + y = tan t.

(f) A nonlinear system of the form x′ = Ax + g(x) can always be linearized about acritical points xc as long as ‖g(x)‖/‖x‖ → 0 when x→ xc .

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations

Example

Indicate if each statement is true or false.

(a) Let f and g be differentiable for every x . If W [f , g ] = 0 for some x , then f and gmust be linearly dependent. False

(b) Suppose we’d like to find a particular solution to y ′′ + y ′ + y = x2 sin x .According to the method of undetermined coefficients we should look for asolution of the form yp = (Ax2 + Bx + C)(D cos x + E sin x). False

(c) Suppose that x1(t) and x2(t) are solutions of x′ = Ax. It is possible forW [x1, x2] = t2 − 4? False

(d) A real 5× 5 matrix cannot have 5 complex eigenvalues. True

(e) Variation of parameters is the only method we’ve learned that can find aparticular solution to y ′′ + y = tan t. False

(f) A nonlinear system of the form x′ = Ax + g(x) can always be linearized about acritical points xc as long as ‖g(x)‖/‖x‖ → 0 when x→ xc . False

(Gary Marple) December 11th, 2017 Math 216: Introduction to Differential Equations