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Differential Equations and Linear Algebra Jason Underdown December 8, 2014 Contents Chapter 1. First Order Equations1 1. Differential Equations and Modeling1 2. Integrals as General and Particular Solutions5 3. Slope Fields and Solution Curves9 4. Separable Equations and Applications 13 5. Linear First{Order Equations 20 6. Application: Salmon Smolt Migration Model 26 7. Homogeneous Equations 28 Chapter 2. Models and Numerical Methods 31 1. Population Models 31 2. Equilibrium Solutions and Stability 34 3. Acceleration{Velocity Models 39 4. Numerical Solutions 41 Chapter 3. Linear Systems and Matrices 45 1. Linear and Homogeneous Equations 45 2. Introduction to Linear Systems 47 3. Matrices and Gaussian Elimination 50 4. Reduced Row{Echelon Matrices 53 5. Matrix Arithmetic and Matrix Equations 53 6. Matrices are Functions 53 7. Inverses of Matrices 57 8. Determinants 58 Chapter 4. Vector Spaces 61 i ii Contents 1. Basics 61 2. Linear Independence 64 3. Vector Subspaces 65 4. Affine Spaces 65 5. Bases and Dimension 66 6. Abstract Vector Spaces 67 Chapter 5. Higher Order Linear Differential Equations 69 1. Homogeneous Differential Equations 69 2. Linear Equations with Constant Coefficients 70 3. Mechanical Vibrations 74 4. The Method of Undetermined Coefficients 76 5. The Method of Variation of Parameters 78 6. Forced Oscillators and Resonance 80 7. Damped Driven Oscillators 84 Chapter 6. Laplace Transforms 87 1. The Laplace Transform 87 2. The Inverse Laplace Transform 92 3. Laplace Transform Method of Solving IVPs 94 4. Switching 101 5. Convolution 102 Chapter 7. Eigenvalues and Eigenvectors 105 1. Introduction to Eigenvalues and Eigenvectors 105 2. Algorithm for Computing Eigenvalues and Eigenvectors 107 Chapter 8. Systems of Differential Equations 109 1. First Order Systems 109 2. Transforming a Linear DE Into a System of First Order DEs 112 3. Complex Eigenvalues and Eigenvectors 113 4. Second Order Systems 115 Chapter 1 First Order Equations 1. Differential Equations and Modeling A differential equation is simply any equation that involves a function, say y(x) and any of its derivatives. For example, (1) y00 = −y: The above equation uses the prime notation (0) to denote the derivative, which has the benefit of resulting in compact equations. However, the prime notation has the drawback that it does not indicate what the independent variable is. By just looking at equation1 you can't tell if the independent variable is x or t or some other variable. That is, we don't know if we're looking for y(x) or y(t). So sometimes we will write our differential equations using the more verbose, but also more clear Leibniz notation. d2y (1) = −y dx2 In the Leibniz notation, the dependent variable, in this case y, always appears in the numerator of the derivative, and the independent variable always appears in the denominator of the derivative. Definition 1.1. The order of a differential equation is the order of the highest derivative that appears in it. So the order of the previous equation is two. The order of the following equation is also two: (2) x(y00)2 = 36(y + x): Even though y00 is squared in the equation, the highest order derivative is still just a second order derivative. 1 2 1. First Order Equations Our primary goal is to solve differential equations. Solving a differential equa- tion requires us to find a function, that satisfies the equation. This simply means that if you replace every occurence of y in the differential equation with the found function, you get a valid equation. There are some similarities between solving differential equations and solving polynomial equations. For example, given a polynomial equation such as 3x2 − 4x = 4; it is easy to verify that x = 2 is a solution to the equation simply by substituting 2 in for x in the equation and checking whether the resulting statement is true. Analogously, it is easy to verify that y(x) = cos x satisfies, or is a solution to equation1 by simply substituting cos x in for y in the equation and then checking if the resulting statement is true. (cos x)00 =? − cos x (− sin x)0 =? − cos x ? − cos x = − cos x X The biggest difference is that in the case of a polynomial equation our solutions took the form of real numbers, but in the differential equation case, our solutions take the form of functions. Example 1.2. Verify that y(x) = x3 − x is a solution of equation2. y00 = 6x ) x(y00)2 = x(6x)2 = 36x3 = 36(y + x) 4 A basic study of differential equations involves two facets. Creating differential equations which encode the behavior of some real life situation. This is called modeling. The other facet is of course developing systematic solution techniques. We will examine both, but we will focus on developing solution techniques. 1.1. Mathematical Modeling. Imagine a large population or colony of bacteria in a petri dish. Suppose we wish to model the growth of bacteria in the dish. How could we go about that? Well, we have to start with some educated guesses or assumptions. Assume that the rate of change of this colony in terms of population is directly proportional to the current number of bacteria. That is to say that a larger popu- lation will produce more offspring than a smaller population during the same time interval. This seems reasonable, since we know that a single bacterium reproduces by splitting into two bacteria, and hence more bacteria will result in more offspring. How do we translate this into symbolic language? (3) ∆P = P ∆t 1. Differential Equations and Modeling 3 This says that the change in a population depends on the size of the population and the length of the time interval over which we make our population measure- ments. So if the time interval is short, then the population change will also be small. Similarly it roughly says that more bacteria correspond to more offspring, and vice versa. But if you look closely, the left hand side of equation3 has units of number of bacteria, while the right hand side has units of number of bacteria times time. The equation can't possibly be correct if the units don't match. However to fix this we can multiply the left hand side by some parameter which has units of time, or we can multiply the right hand side by some parameter which has units of 1/time. Let's multiply the right hand side by a parameter k which has units of 1/time. Then our equation becomes: (4) ∆P = kP ∆t Dividing both sides of the equation by ∆t and taking the limit as ∆t goes to zero, we get: ∆P dP lim = = kP ∆t!0 ∆t dt dP (5) = kP dt Here k is a constant of proportionality, a real number which allows us to balance the units on both sides of the equation and it also affords some freedom. In essence it allows us to defer saying how closely P and its derivative are related. If k is a large positive number, then that would imply a large rate of change, and a small positive number greater than zero but less than one would be a small rate of change. If k is negative then that would imply the population is shrinking in number. Example 1.3. If we let P (t) = Cekt, then a simple differentiation reveals that this is a solution to our population model in equation5. Suppose that at time 0, there are 1000 bacteria in the dish. After one hour the population doubles to 2000. This data corresponds to the following two equations which allow us to solve for both C and k: 1000 = P (0) = Ce0 = C =) C = 1000 2000 = P (1) = Cek The second equation implies 2000 = 1000ek which is equivalent to 2 = ek which is equivalent to k = ln 2. Thus we see that with these two bits of data we now know: P (t) = 1000eln(2)·t = 1000(eln(2))t = 1000 · 2t This agrees exactly with our knowledge that bacteria multiply by splitting into two. 4 4 1. First Order Equations 1.2. Linear vs. Nonlinear. As you may have surmised we will not be able to exactly solve every differential equation that you can imagine. So it will be important to recognize which equations we can solve and those which we can't. It turns out that a certain class of equations called linear equations are very amenable to several solution techniques and will always have a solution (under modest assumptions), whereas the complementary set of nonlinear equations are not always solvable. A linear differential equation is any differential equation where solution func- tions can be summed or scaled to get new solutions. Stated precisely, we mean: Definition 1.4. A differential equation is linear is equivalent to saying: If y1(x) and y2(x) are any solutions to the differential equation, and c is any scalar (real) number, then (1) y1(x) + y2(x) will be a solution and, (2) cy1(x) will be a solution. This is a working definition, which we will change later. We will use it for now because it is simple to remember and does capture the essence of linearity, but we will see later on that we can make the definition more inclusive. That is to say that there are linear differential equations which don't satisfy our current definition until after a certain piece of the equation has been removed. Example 1.5. Show that y1(x) + y2(x) is a solution to equation1 when y1(x) = cos x and y2(x) = sin x.
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