MAT 275: Modern Differential Equations Lecture Notes

MAT 275: Modern Differential Equations Lecture Notes

MAT 275: Modern Differential Equations Lecture Notes Jeremiah Jones Contents 1 First-Order Equations 3 1.1 Introduction . 3 1.1.1 Definition of a Differential Equation . 3 1.1.2 Differential Equations as Physical Laws . 3 1.1.3 Differential Equations as Models . 5 1.1.4 Solutions to Some Differential Equations . 5 1.1.5 Verifying Solutions . 7 1.1.6 Classifying Differential Equations . 8 1.1.7 Equilibrium Solutions and Asymptotic Behavior . 9 1.2 Separable Equations . 9 1.3 First Order Linear Equations . 13 1.4 Modeling with First-Order Equations . 17 1.4.1 Radioactive Decay . 17 1.4.2 Newton’s Law of Cooling . 18 1.4.3 Solution Mixing . 19 1.4.4 Population Dynamics . 21 1.5 Exact Equations . 22 1.5.1 Partial Derivatives . 23 1.5.2 Differentials . 24 1.5.3 Exactness Criteria . 24 1.5.4 Solving Exact Equations . 26 1.5.5 Exact Integrating Factors . 28 1.6 Numerical Methods . 30 1.6.1 Forward Euler Method . 31 1.6.2 Backward Euler Method . 32 1.6.3 The Improved Euler Method . 34 2 Second-Order Equations 36 2.1 Introduction to second-order linear equations . 36 2.2 Second-order linear homogeneous differential equations with constant coefficients . 36 2.2.1 Fundamental Solutions . 37 2.2.2 Forming a General Solution . 38 2.2.3 Examples . 38 2.3 Solutions of second-order linear equations, linear independence and the Wronskian . 41 2.3.1 Operator Notation . 41 2.3.2 Existence of a unique solution . 42 2.3.3 Constant Solutions . 43 1 2.3.4 Linear Superposition and Linear Independence . 43 2.3.5 Complex Solutions . 44 2.3.6 Converting to a system of first-order equations . 45 2.4 Complex roots of the characteristic equation . 46 2.4.1 Euler’s Formula . 47 2.4.2 The General Solution . 49 2.4.3 Examples . 50 2.5 Repeated roots of the characteristic equation . 52 2.5.1 Reduction of order . 52 2.5.2 Finding the second solution . 53 2.5.3 Examples . 54 2.6 Summary of the Characteristic Solutions . 55 2.7 Higher-order equations . 56 2.7.1 Examples . 56 2.8 Non-homogeneous second-order linear differential equations with constant coefficients 57 2.8.1 The Particular Solution . 58 2.8.2 Method of Undetermined Coefficients . 59 2.9 The Mass-Spring System . 65 2.9.1 Undamped Free Oscillations . 67 2.9.2 Damped Free Oscillations . 69 2.10 LRC Series Circuits . 71 2.11 Forced Oscillations . 72 2.11.1 Forced Oscillations with damping . 73 2.11.2 Forced oscillations with no damping . 75 2.11.3 Beat Phenomenon . 77 2.11.4 LRC Series Circuits . 77 2.12 The Nonlinear Pendulum . 78 3 Laplace Transforms 80 3.1 Introduction . 80 3.1.1 Integral Transforms . 80 3.1.2 The Laplace Transform Definition . 81 3.1.3 Examples . 81 3.1.4 Properties of the Laplace Transform . 82 3.1.5 Further Examples . 84 3.1.6 Inversion of the Laplace Transform . 85 3.1.7 Table of Results . 88 3.2 Solving IVPs with Laplace Transforms . 89 3.3 Laplace Transforms of Piecewise Functions . 94 3.3.1 Step Functions and Piecewise Functions . 94 3.3.2 IVPs with Piecewise Functions . 97 3.4 The Impulse Function and its Laplace Transform . 102 3.4.1 Definition of the Impulse Function . 102 3.4.2 IVPs with Impulse Functions . 104 3.5 The Convolution Formula . 107 2 Chapter 1 First-Order Equations 1.1 Introduction 1.1.1 Definition of a Differential Equation A differential equation is an equation that involves a function y(x) and some of its derivatives. The order of a differential equation is the highest derivative that appears in the equation. Some examples of differential equations are y0 + 2xy − 1 = 0 y0 − 3y = 0 1 yy0 − = 0 x y00 − 3y2 + ex = 0 y00 + 4 sin(y) = 0 Each of the above equations gives some relationship between a function y(x) and its derivatives. Solving a differential equation consists of finding all of the functions y that satisfy the equation. In general, differential equations have infinitely many solutions and extra information, referred to as initial conditions, is required in order to find a unique solution. This is related to the fact that integrating a function always results in adding an arbitrary constant. Initial conditions tell us the value of the function or one of its derivatives at a certain point. A differential equation together with initial conditions is called an initial value problem. The general form of an nth order differential equation can be stated as F (x, y, y0, y00, ..., y(n)) = 0. Each of the examples stated earlier are written in this form. The order of a differential equation is equal to the number of initial conditions needed to determine a unique solution. For example, first order equation require one initial condition, second order equations require two, etc. 1.1.2 Differential Equations as Physical Laws Differential equations are important to scientists and engineers because physical laws usually take the form of a differential equation. Some examples are Newton’s laws of motion, conservation of 3 mass, momentum and energy, Newton’s law of cooling, Kepler’s laws of planetary motion, etc. We will take a look at some of these more closely throughout the course. Newton’s 2nd Law Let y(t) denotes the position of an object with mass m at time t and F denote the total force acting on it. Then Newton’s second law takes the form a 2nd order differential equation for y(t): F = ma(t) = mv0(t) = my00(t). This is usually referred to as the equation of motion. In general, the force F can be a function of y and t. Example 1.1.1. For the case of a falling object with no air resistance, F = mg, which results in Z 0 0 mv = mg ⇒ v = g ⇒ v(t) = gdt = v0 + gt Z 1 y(t) = v(t)dt = gt2 + v t + y 2 0 0 where v0 and y0 are the initial velocity and initial height. Example 1.1.2. Now consider a falling object with air resistance, i.e., a drag force, that is directly proportional to the velocity. There are two forces acting on the mass: gravity and drag. Assuming that down is the positive direction, the total force would be F = mg − γv where g = 9.8m/s2 is the acceleration due to gravity and γ is the drag coefficient. Newton’s 2nd law then tells us mg − γv = mv0(t). Newton’s Law of Cooling Let y(t) denote the temperature of an object that is placed in a surrounding at temperature T . Then Newton’s Law of Cooling gives us a differential equation for y(t): dy = k(T − y) dt where k > 0 is a constant called the heat transfer coefficient. The value of the constant k is determined by the thermal properties of the object. Note that if the object is hotter than its surroundings then y > T ⇒ T − y < 0 ⇒ y0 < 0. In other words, y is decreasing (giving off heat to the surroundings). On the other hand, if y is colder than T , then y0 > 0 and y is increasing. In either case, the object eventually comes into thermal equilibrium with the surroundings, i.e., y(t) → T as t → ∞. 4 1.1.3 Differential Equations as Models In the last two examples, we were given problems that have physical laws that govern them but when studying more complicated systems, we must use mathematical models. A mathematical model is a set of equations that attempts to describe a system. Modeling usually relies on logic, intuition and empirical data. For example, if we wanted to do a mathematical study of population growth, there are no physical laws we can use so we must develop a model. The Logistic Model One particular model for population growth is called the logistic model and takes the form of the differential equation y y0 = ry 1 − K where y(t) is the population at time t , r is the growth rate and K is the maximum number of individuals the population can sustain. We will study this model in depth later in the course. 1.1.4 Solutions to Some Differential Equations The easiest type of differential equations are explicit equations where y0(x) or y00(x) is given ex- plicitly in terms of x. You have already encountered these in Calculus I and they only involve integration. Example 1.1.3. Solve the differential equation dy = x2 − ex dx Solution: Multiplying through by dx and integrating each side results in Z Z 1 dy = x2 − ex dx ⇒ y + C = x3 − ex + C . 1 3 2 Therefore, we can state the final solution as 1 y(x) = x3 − ex + C − C . 3 2 1 If we define C = C2 − C1, then we have 1 y(x) = x3 − ex + C. 3 As a general rule, we can always combine two constants this way so whenever integrating both sides of an equation, we only need to add a constant to one side. Notice that since C can have any value, we have infinitely many solutions. Example 1.1.4. Solve the differential equation y0 = y. 5 Solution: First rewrite the equation in differential form as dy = y. dx Multiplying each.

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