Lecture 1. Introduction1. What Is A Differential Equation?2. Why do we care?3. Examples4. Basic Definitions5. General StrategyDifferential EquationsLECTURE 1Introduction1. What Is A Differential Equation?A differential equation is an equation that contains derivatives of a function. On one hand, thisseems obvious. On the other, it’s a remarkably subtle idea.Why? Let’s think back to algebra. Equations there were of the formf(x) = cfor some known function f(x) and some constant c. f(x) may have been the derivative of anotherfunction, but we’d explicitly calculated it...the function was never unknown. Solving the equationmeant finding values of the independant variable x that caused the function to have the desiredvalue. So far, so good; this is old news. But what does that have to do with anything at themoment?Here’s the difference: the unknowns in a differential equation are functions, not numbers. Thisis an important conceptual jump, the central one in this course. We don’t plug in a value for xand see what f(x) evalutes to; we plug in expressions for a function and see what the whole messsimplifies to.Another way of putting it (which you don’t have to especially worry about, but which can behelpful) is that a differential equation has the formL[y] = g(x)where y(x) is an unknown function (a variable in this case), g(x) is some other (known) functionnot involving y, but rather only the independent variable, and L is a function defined on functions(what we in mathematics call an “operator”).An example of an operator is justddx. If you plug in a function y, you getdydx, which is justanother function.This has several consequences. First, it means that differential equations are hard. For thisclass, by “hard,” I mean “they take some getting used to.” We’re used to our equations havingnumerical solutions, and we understand exactly what an equation says when we look at it. Now,we’ll need to learn how to allow ourselves to abstract our notion of “equation.” In the real world,differential equations are actually difficult. Most can’t be solved, and some that can be solvedrequire computers to do so.Second, differential equations, when they have solutions, have infinitely many. Why? Well,let’s look at a differential equation you’ve all seen before.dydx= x (1.1)This is just an equation from calculus. What we want to find is a function y(x) that, whendifferentiated at a point x gives us just x. We know how to solve this: just integrate.y(x) =x22+ c (1.2)1Differential Equations Lecture 1: IntroductionSince differentiation ignores constants, when we integrate we have to account for this by addingin the constant of integration c. This is where the infinity of solutions comes from, and this willcome up every time we solve a differential equation. A solution of the form 1.2 is called a generalsolution because it can express every possible solution of the differential equation.Usually, though, we’re interested in solving a particular initial value problem. Initial valueproblems are a combination of a differential equation and some initial conditions which fix a specific,or particular solution. They have the formy(k)(t0) = ykfor whichever derivatives of y are necessary. This is no different from an integration problem likeEquation 1.1; we fixed a particular value of y to determine c, and that gave us a unique function.So, when we are faced with an initial value problem, we’re doing two things: we’re finding allfunctions that satisfy the equation and then choosing the particular one that satisfies the giveninitial conditions.Let’s say we’ve got a differential equation. A function y is a solution to the differential equationif, when we take the relevant derivatives and plug them in, the equation “works”: everythingsimplifies the way it should.Example 1.1. Consider2x2y00− 3xy0− 3y = 0.I claim y(x) = x−12is a solution to this equation. To check this, we’ll need the first and secondderivatives:y0(x) = −12x−32y00(x) =34x−52Then:2x234x−52− 3x−12x−32− 3x−12= 032x−12+32x−12− 3x−12= 00 = 0So y(x) = x−12is in fact a solution. Here are some others:y(x) = x3,y(x) = 12x−12,y(x) = 6x−12− 2x3.In fact, every solution to this differential equation is of the formy(x) = c1x−12+ c2x3(1.3)for some constants c1and c2. So Equation 1.3 is the general solution to the differential equation,and we could pick out a particular solution if we specified some initial conditions. We refer to a solution such as in Example 1.1 as an explicit solution since it is of the formy = f(x); in other words, the only place y appears is on the left hand side and it is just by itself,2Differential Equations Lecture 1: Introductionraised to the first power. We’ll see examples soon where we won’t be able to solve for y. In thosecases, we will have to be content to find implicit solutions.Example 1.2. y2= t2− 3 is an implicit solution toy0=tyy(−2) = 1. Exercise. Check that, indeed, y2= t2− 3 is a solution to the initial value problem given inExample 1.2.Example 1.3. Let’s find the explicit solution to the problem given in Example 1.2. We knowthat y2= t2− 3 is a solution; now we want to solve for y. Doing so yields y = ±√t2− 3. This isa problem: we want one expression for y, but here we have two. To figure out which one we want,we use the initial condition y(−2) = 1. Only one of the two possibilities will satisfy the initialcondition. Plugging in, we see that the positive one is correct, andy(t) =pt2− 3is the explicit solution to thise initial value problem. Checking that things are solutions is very important, especially early on. It’s a good habit toget into, because it’s very easy to make a sloppy mistake (such as a sign error) that can drasticallychange the solution.The other issue we need to keep in mind is that solutions to differential equations are generallyonly valid on certain intervals. Consider Example 1.1. We know y(x) = x−12is a solution to thisequation. . . but for what values of x? It can’t be all of them, since y(x) isn’t even defined at x = 0.It turns out that this solution is valid for x > 0. In fact, looking at Equation 1.3, no solution to thisdifferential equation exists at x = 0! So even though, symbolically, y(x) satisfied the differentialequation for all x, the solution implied a certain restriction on the independent variable.This is generally going to be the case: it’s very
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