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

Transcription

1 Week #10 : Systems of DEs Goals: Introduction to Systems of Differential Equations Solving systems with distinct real eigenvalues and eigenvectors 1

2 Systems of Differential Equations - Introdution - 1 Systems of Differential Equations - Introdution We have gone about as far as we can with interesting single-variable DEs. In practice, more complex systems involve multiple, interrelated 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 in an earthquake.

3 Systems of Differential Equations - Introdution - 2 Demonstration

4 Model: Spring Models for Buildings - 1 Model: Spring Models for Buildings To understand the dynamics in a complex system, we need to go back to basics. Problem. Draw a double-spring diagram, and a double-column diagram. If both x 1 and x 2 are increased by an equal amount, what is the force in the second spring/column?

5 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 a differential equation for the first mass. Repeat for the second mass.

6 Model: Spring Models for Buildings - 3 Problem. Write out the system of differential equations obtained.

7 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-order system 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. c 1, λ. Problem. Define a new vector of 4 variables, w, that will allow the conversion of the higher-order system to a first-order system.

8 Converting Higher-Order DEs to 1st Order Systems - 2 Define the derivative of w in terms of w itself, making use of the DE as necessary. This is now a first-order system of differential equations. (The variables we are most interested in for this example are w 1 = x 1, and w 3 = x 2, the positions of each mass.)

9 Problem. Put the equations into matrix format. Converting Higher-Order DEs to 1st Order Systems - 3

10 Converting Higher-Order DEs to 1st Order Systems - 4 Problem. Use a technique from earlier in the course to find a form of the solution.

11 Eigenvalues In Solutions - 1 Solving w = A w, assume w = ve λt We have reduced our solving of the first-order system of equations to finding the eigenvalues and eigenvectors of a matrix. Problem. If we found eigenvalues λ = 2, 3, 4, 5, what form would the solution take?

12 Eigenvalues In Solutions - 2 Solving w = A w, assume w = ve λt Problem. If we found eigenvalues λ = 1 ± 2i, 2 ± 3i, what form would the solution take?

13 Eigenvalues In Solutions - 3 Solving w = A w, assume w = ve λt Problem. If λ = 1 ± 2i, 2 ± 3i were the values for our building model in the demonstration software, what would be dangerous frequencies for an external force and why?

14 Matrices and Linear Systems - Homogeneous Theory - 1 Matrices and Linear Systems - Homogeneous Theory How does linear algebra help solve systems of linear differential equations? Consider the system of differential equations x 1 (t) = 3x 1(t) 4x 2 (t) and x 2 (t) = 4x 1(t) 7x 2 (t) Problem. Write this system of equations out in matrix form.

15 Matrices and Linear Systems - Homogeneous Theory - 2 In the matrix/vector form, what kind of rules about the solution can we rely on? Theorem. Let A(t) and f(t) be continuous on an open interval I. If t 0 I and u R n, then there exists a unique solution x(t) on I to x (t) = A(t) x(t) + f(t) where x(t 0 ) = x 0. Theorem. Let A(t) be a continuous (n n)-matrix on an open interval I. If x 1 (t),..., x n (t) are linearly independent solutions to the homogenous system x (t) = A(t) x(t) on I, then every solution has the form x(t) = c 1 x 1 (t) + + c n x n (t).

16 Verifying Solutions - Example - 1 Problem. Consider the system of equations x (t) = x(t) Show that the following vector-valued functions are solutions to the system. (a) x 1 = e 2t e 2t e 2t

17 Verifying Solutions - Example - 2 x (t) = x(t) (b) x 2 = e t 0 e t (c) x 3 = 0 e t e t

18 with x 1 = x (t) = e 2t e 2t, x 2 = e 2t x(t) e t 0 e t, x 3 = 0 e t e t Verifying Solutions - Example - 3 Problem. Based on the earlier results, write out the general solution to the system of equations.

19 Matrices and Linear Systems - Non-Homogeneous Theory - 1 Matrices and Linear Systems - Non-Homogeneous Theory In analogy to our earlier work, what if we have a system which is non-homogeneous? The standard form for a linear system with a non-homogeneous component is: x (t) = A(t) x(t) + f(t)

20 Matrices and Linear Systems - Non-Homogeneous Theory - 2 Theorem. Let A(t) be a continuous (n n)-matrix on an open interval I and let x 1 (t),..., x n (t) be linearly independent solutions to If x NH (t) satisfies x (t) = A(t) x(t) on I. x (t) = A(t) x(t) + f(t) on I, then every solution of the nonhomogeneous system has the form x(t) = c 1 x 1 (t) + + c n x n (t) + x NH (t).

21 Matrices and Linear Systems - Non-Homogeneous Theory - 3 Problem. Verify that x NH (t) = [t 1, t, t + 1] t is a particular solution to x (t) = x(t) + 2t

22 Matrices and Linear Systems - Non-Homogeneous Theory - 4 Problem. Find the general solution to x (t) = x(t) t 1 2 x NH (t) = t 1 t, x 1 = t + 1 e 2t e 2t, x 2 = e 2t e t 0 e t, x 3 = 0 e t e t

23 DE Systems with Constant Coefficients - 1 DE Systems with Constant Coefficients Knowing how we can combine individual solutions, how do we find those solutions in the first place? We saw earlier that eigenvalues and eigenvectors are tools that could assist us. Theorem. Let A be a constant (n n)-matrix with n linearly independent eigenvectors u 1,..., u n. If r i is the eigenvalue corresponding to u i, then the general solution to x = A x is x(t) = c 1 e r 1t u 1 + c 2 e r 2t u c n e r nt u n.

24 DE Systems with Constant Coefficients - 2 Problem. Solve x (t) = A x(t) where A = [ ]

25 DE Systems with Constant Coefficients - 3 x (t) = A x(t) where A = [ ]

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28 DE Systems with Constant Coefficients - 4 Problem. Verify your solution is correct. x (t) = A x(t) where A = [ ]

29 Problem. Solve x (t) = A x(t) where A := and x(0) = Real Eigenvalue Solutions - Example

30 Real Eigenvalue Solutions - Example - 2 A := and x(0) =

31 A := Real Eigenvalue Solutions - Example and x(0) =

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