Introduction What is a Differential Equation? A differential equation (DE) is an equation that relates a function (usually unknown) to its own derivatives. Example 1: The equation + y3 unknown function, y = f(x), and one of its derivatives,. = x 2 is a differential equation since it includes an Example 2: Another example of a differential equation (a more complicated one) is ( ) d 3 2 y + 3 et ty = cos t. When solving equations in an algebra course, you typically start with an equation such as x 2 6x = 8 in which the goal is to find all (real or complex) numbers of the variable (x, in this case) such that the equation holds. On the other hand, to solve a differential equation means to determine all possible functions that satisfy the equation. Example 3: Consider the differential equation + 2y = 1. Claim: A solution of this DE is y = 1 2 + Ce 2x, where C is an arbitrary constant. We can verify that this is a solution simply by plugging it back into the equation: Therefore, y = 1 2 + Ce 2x = = 2Ce 2x. ( ) 1 + 2y = 1 = 2Ce 2x + 2 2 + Ce 2x = 1 = 2Ce 2x + 1 + 2Ce 2x = 1 = 1 = 1 which is a true statement. In this course, we will learn of several different techniques used to analytically solve certain types of differential equations. 1
Why Do We Stu Differential Equations? ˆ DEs have many applications (e.g. in physics, chemistry, engineering, finance, and the biological sciences, just to name a few). ˆ DEs describe and capture how things move, change, and evolve. For example, the motion of the earth around the sun is described by a DE. The way species interact with each other (e.g. mice and owls) is also described by a DE. The way temperature changes over time (in a particular space or through some object) is described by a DE. In problems involving population growth, it is often assumed that the growth rate is proportional to the size of the population. In other words, if P (t) represents the population at time t and P (t) > 0, then P (t) = rp (t), where r is the proportionality constant, is the differential equation that describes how the population grows over time. (r is also called the growth rate.) The population function P (t) that satisfies the DE is P (t) = P 0 e rt, where P 0 = P (0) represents the population at t = 0 (usually called the initial population). This tells us that the population grows exponentially over time. 2
Section 1.1: Some Basic Mathematical Models; Direction Fields Mathematical Modeling Definition 1: A mathematical model is a differential equation that describes some physical process. Example 1: The DE dp = rp, where dp > 0 and r > 0, models population growth. Note: This is an accurate model for populations with unlimited resources. Realistically, r is not necessarily constant. The assumption that r is constant actually works for small populations. If we consider what happens in the long term, we must take into account the fact that resources of food and space are limited. A better model would be dp = r ( 1 P K where r and K are constants (K is also known as the carrying capacity of the environment). This differential equation is commonly called the logistic equation. Example 2: Newton s Law of Cooling This law is used to model the change in temperature of an object (with some temperature at time t) placed in an environment having a different temperature. For example, think about how the temperature of a coffee mug filled with hot coffee changes when the mug is placed in a refrigerator. Newton s Law of Cooling states that dt = k(t R) where T is the temperature of the object at time t, R is the temperature of the surrounding environment (constant), and k is the constant of proportionality. Example 3: Spread of a Disease Suppose that we analyze the spread of a contagious disease, such as a flu virus. It is assumed that the rate at which the virus spreads is jointly proportional to the number of people who have contracted the disease (call this number x(t)) and the number of people who have not been exposed to the disease (call this number y(t)). Mathematically, this means that = kxy where is the rate at which the disease spreads and k is the constant of proportionality. ) P 3
Example 4: Falling Bo Suppose that an object with mass m is thrown upward from the roof of a building. If s(t) represents the object s position from the ground at time t, then the object s acceleration is d2 s. Let the upward direction be positive and assume that the force of gravity is the 2 only force that acts on the object. Then by Newton s second law of motion (i.e. the net force on the object is proportional to the object s acceleration where m is the constant of proportionality) yields the differential equation m d2 s = mg = d2 s 2 = g. 2 where g is the acceleration due to gravity at the earth s surface. Notice the minus sign in the equation. This is due to the fact that the weight, mg, of the object is a force directed downward which is opposite of the positive direction. Example 5: Series Circuits Consider a single-loop series circuit consisting of a resistor, an inductor, and a capacitor. Let i(t) represent the current in the circuit after a switch is closed and let q(t) represent the charge on the capacitor at time t. Note that R denotes resistance, L denotes inductance, and C denotes capacitance. According to Kirchoff s second law, the impressed voltage E(t) on a closed loop equals the sum of the voltage drops in the loop. Since i(t) is related to q(t) on the capacitor by i = dq/, we add the following voltage drops inductor = L di = d2 q 2 resistor = ir capacitor = 1 C q = R dq and equate the sum to E(t) to obtain the differential equation L d2 q 2 + Rdq + 1 C q = E(t). Figure 1: An RLC series circuit. 4
Direction Fields (Slope Fields) Before we learn how to solve differential equations analytically, let s stu what happens geometrically when we solve a differential equation in which the only derivative that appears is the first derivative. Example 6: Consider the differential equation = x. Let s focus on the actual meaning y of this equation. If a solution exists and passes through a point (x 0, y 0 ) in the xy-plane, then the DE gives the derivative at that point. Thus, at (x 0, y 0 ), we can draw a tangent vector which indicates the direction of the solution curve at that point. The following table lists several selected points and the derivative at each point. The figure next to the table shows the tangent vectors placed at their designated points. Two different solution curves, which happen to be semicircles, are also shown. x y 1 1 1 0 1 0 1 1 1 1 0 ± 1 1 1 0 1 0 1 1 1 1 0 ± Figure 2: Tangent vectors at specific points. Definition 2: A direction field is a collection of tangent vectors through all points in the xy-plane. Direction fields allow us to visualize the solution curve through a specific point even if we cannot analytically determine the form of the solution. Since drawing tangent vectors by hand can be a very tedious task, we usually rely on numerical software (such as MAPLE) to sketch direction fields. 5
Example 7: The following is a direction field (generated by MAPLE) for the differential equation in Example 6. MAPLE Commands: > with(plots): > fieldplot([1,-x/y],x=-4..4,y=-4..4, arrows=line, fieldstrength=fixed(0.5), grid=[17,17]); Example 8: The following is a direction field (generated by MAPLE) for the differential equation = cos y cos t. 6
MAPLE Commands: > with(detools): > eq:= diff(y(t),t) = cos(y(t))-cos(t); > DEplot(eq,y(t),t=-6..6,y=-6..6); Example 9: The following is a direction field for the differential equation = y(4 y). MAPLE Commands: > with(detools): > eq:= diff(y(t),t) = y(t)*(4-y(t)); > DEplot(eq,y(t),t=-1..1,y=-1..5,color=black); 7
Isoclines A more efficient way to sketch direction fields by hand is to use what are called isoclines. Definition 3: An isocline is a set of points in a direction field for which there is a constant c such that = c at these points. Therefore, for a differential equation = f(x, y), any member of the family f(x, y) = c is an isocline, which literally means a curve along which the inclination of the tangents is the same. Example 10: Consider the differential equation = y = c Here, f(x, y) = y. Thus, y = c represents a one-parameter family of horizontal lines. Example 11: Consider the differential equation in Example 6: Then, = x y x y = c = y = 1 c x (isoclines) Note that y = 1 x is a one-parameter family of lines that pass through the origin and have c slope 1. For c = 1, the isocline is y = x. This means that every point on the line y = x c must have a tangent vector with slope 1 (since = c = 1). For c = 1, the isocline is y = x, and thus, every point on the line y = x must have a tangent vector with slope 1. Note: Isoclines are not necessarily solution curves. Definition 4: An isocline corresponding to c = 0 is called a nullcline. Example 12: In Example 11, the line x = 0 is a nullcline. 8