week #10 : systems of des goals: introduction to systems of di … · 2016-06-27 · systems of di...

Week #10 : Systems of DEs

Goals:

• Introduction to Systems of Differential Equations• Solving systems with distinct real eigenvalues and eigenvectors

1

Systems of Differential Equations - Introdution - 1

Systems of Differential Equations - Introdution

We have gone about as far as we can with interesting single-variableDEs. In practice, more complex systems involve multiple, inter-related variables.

• Complex physical and electronic systems• Interacting populations like predator/prey and host/parasites

One particularly visceral model is that of a multi-story building inan earthquake.

Systems of Differential Equations - Introdution - 2

Demonstration

Model: Spring Models for Buildings - 1

Model: Spring Models for Buildings

To understand the dynamics in a complex system, we need to go backto basics.

Problem. Draw a double-spring diagram, and a double-column di-agram.

If both x1 and x2 are increased by an equal amount, what is the forcein the second spring/column?

Model: Spring Models for Buildings - 2

We will use the spring system as our model for the force calculations,simply because it is more familiar, and easier to visualize.

Problem. Draw a free-body diagram for the first mass. Obtain adifferential equation for the first mass.

Repeat for the second mass.

Model: Spring Models for Buildings - 3

Problem. Write out the system of differential equations ob-tained.

Converting Higher-Order DEs to 1st Order Systems - 1

Converting Higher-Order DEs to 1st Order Systems

For consistency of analysis, we will transform this second-ordersystem into a larger first-order system.

Notation: In this section of the course:

• vectors with be written with vector hats, e.g. ~x,•matrices will be written using capital letters, e.g. A or M , and• scalars will be in lower-case, e.g. c1, λ.

Problem. Define a new vector of 4 variables, ~w, that will allow theconversion of the higher-order system to a first-order system.

Converting Higher-Order DEs to 1st Order Systems - 2

Define the derivative of ~w in terms of ~w itself, making use of the DEas necessary.

This is now a first-order system of differential equations.(The variables we are most interested in for this example are w1 = x1,and w3 = x2, the positions of each mass.)

Converting Higher-Order DEs to 1st Order Systems - 3

Problem. Put the equations into matrix format.

Converting Higher-Order DEs to 1st Order Systems - 4

Problem. Use a technique from earlier in the course to find a formof the solution.

Eigenvalues In Solutions - 1

Solving ~w ′ = A~w, assume ~w = ~veλt

We have reduced our solving of the first-order system of equations tofinding the eigenvalues and eigenvectors of a matrix.

Problem. If we found eigenvalues λ = 2, 3, 4, 5, what form wouldthe solution take?

Eigenvalues In Solutions - 2

Solving ~w ′ = A~w, assume ~w = ~veλt

Problem. If we found eigenvalues λ = −1± 2i,−2± 3i, what formwould the solution take?

Eigenvalues In Solutions - 3

Solving ~w ′ = A~w, assume ~w = ~veλt

Problem. If λ = −1 ± 2i,−2 ± 3i were the values for our build-ing model in the demonstration software, what would be dangerousfrequencies for an external force and why?

Matrices and Linear Systems - Homogeneous Theory - 1

Matrices and Linear Systems - Homogeneous Theory

How does linear algebra help solve systems of linear differential equa-tions?Consider the system of differential equations

x′1(t) = 3x1(t)− 4x2(t) and

x′2(t) = 4x1(t)− 7x2(t)

Problem. Write this system of equations out in matrix form.

Matrices and Linear Systems - Homogeneous Theory - 2

In the matrix/vector form, what kind of rules about the solution canwe rely on?

Theorem. Let A(t) and ~f (t) be continuous on an open intervalI. If t0 ∈ I and u ∈ Rn, then there exists a unique solution ~x(t)

on I to ~x′(t) = A(t)~x(t) + ~f (t) where ~x(t0) = ~x0.

Theorem. Let A(t) be a continuous (n × n)-matrix on an openinterval I. If ~x1(t), . . . , ~xn(t) are linearly independent solutionsto the homogenous system ~x ′(t) = A(t)~x(t) on I, then everysolution has the form

~x(t) = c1~x1(t) + · · · + cn~xn(t).

Verifying Solutions - Example - 1

Problem. Consider the system of equations

~x ′(t) =

0 1 11 0 11 1 0

~x(t)

Show that the following vector-valued functions are solutions to thesystem.

(a) ~x1 =

e2t

e2t

e2t

Verifying Solutions - Example - 2

~x ′(t) =

0 1 11 0 11 1 0

~x(t)

(b) ~x2 =

−e−t0e−t

(c) ~x3 =

0e−t

−e−t

Verifying Solutions - Example - 3

~x ′(t) =

0 1 11 0 11 1 0

~x(t)

with ~x1 =

e2t

e2t

e2t

, ~x2 =

−e−t0e−t

, ~x3 =

0e−t

−e−t

Problem. Based on the earlier results, write out the general solu-tion to the system of equations.

Matrices and Linear Systems - Non-Homogeneous Theory - 1

Matrices and Linear Systems - Non-Homogeneous The-ory

In analogy to our earlier work, what if we have a system which isnon-homogeneous?

The standard form for a linear system with a non-homogeneous com-ponent is:

~x ′(t) = A(t)~x(t) + ~f (t)

Matrices and Linear Systems - Non-Homogeneous Theory - 2

Theorem. Let A(t) be a continuous (n × n)-matrix on an openinterval I and let ~x1(t), . . . , ~xn(t) be linearly independent solu-tions to

~x ′(t) = A(t)~x(t) on I.

If ~xNH(t) satisfies

~x ′(t) = A(t)~x(t) + ~f (t) on I,

then every solution of the nonhomogeneous system has the form

~x(t) = c1~x1(t) + · · · + cn~xn(t) + ~xNH(t).

Matrices and Linear Systems - Non-Homogeneous Theory - 3

Problem. Verify that ~xNH(t) = [t− 1, −t, t+ 1]t is a particularsolution to

~x ′(t) =

0 1 11 0 11 1 0

~x(t) +

0−2t− 1

2

Matrices and Linear Systems - Non-Homogeneous Theory - 4

Problem. Find the general solution to

~x ′(t) =

0 1 11 0 11 1 0

~x(t) +

0−2t− 1

2

~xNH(t) =

t− 1−tt + 1

, ~x1 =

e2t

e2t

e2t

, ~x2 =

−e−t0e−t

, ~x3 =

0e−t

−e−t

DE Systems with Constant Coefficients - 1

DE Systems with Constant Coefficients

Knowing how we can combine individual solutions, how do we findthose solutions in the first place?We saw earlier that eigenvalues and eigenvectors are tools thatcould assist us.

Theorem. Let A be a constant (n × n)-matrix with n linearlyindependent eigenvectors ~u1, . . . , ~un.

If ri is the eigenvalue corresponding to ~ui,

then the general solution to ~x ′ = A~x is

~x(t) = c1 er1t ~u1 + c2 e

r2t ~u2 + · · · + cn ernt ~un .

DE Systems with Constant Coefficients - 2

Problem. Solve ~x ′(t) = A~x(t) where A =

[2 −31 −2

].

DE Systems with Constant Coefficients - 3

~x ′(t) = A~x(t) where A =

[2 −31 −2

].

DE Systems with Constant Coefficients - 4

~x ′(t) = A~x(t) where A =

[2 −31 −2

].

Problem. Verify your solution is correct.

Real Eigenvalue Solutions - Example - 1

Problem. Solve ~x ′(t) = A~x(t) where

A :=

1 2 −11 0 14 −4 5

and ~x(0) =

−100

.

Real Eigenvalue Solutions - Example - 2

A :=

1 2 −11 0 14 −4 5

and ~x(0) =

−100

.

Real Eigenvalue Solutions - Example - 3

A :=

1 2 −11 0 14 −4 5

and ~x(0) =

−100

.