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Differential Equations: Order, Solutions, and IVPs
Calculus 2 · Chapter 5: Differential EquationsSection 5.1 · lesson 22 of 5411 min · free
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In this lesson
Order, solutions, and initial-value problems.
- See why the unknown is a **function**, not a number
- Classify by order, and distinguish ODEs from PDEs
- Separate the general solution family from a particular member
- Use initial conditions to pin down the constants
Transcript
Everything said in the video, section by section (1,656 words). Read along, or search it.
MAT-265 - Calculus 2 · Section 5.1
Introduction to Differential Equations
When a forensic investigator estimates time of death. When a cup of coffee cools on a desk. When a population of bacteria multiplies in a dish, or a charged capacitor drains through a circuit. In every one of those situations, the underlying rule is never a statement about a quantity's value. It's a statement about its rate of change. Newton's law of cooling doesn't say how hot the coffee is. It says the coffee cools at a rate proportional to how much hotter it is than the room. Such a rule, written in the language of derivatives, is a differential equation. And because so many natural laws are fundamentally about rates, differential equations are arguably the single most important application of calculus. They're the bridge from the abstract machinery of derivatives, to the concrete prediction of how systems evolve over time.
The Idea
The Equation That Describes Change Itself
So let's be precise about what one is. A differential equation is an equation that relates a function to one or more of its derivatives. And here's the shift that defines the whole subject. An algebraic equation like x squared minus four equals zero is solved by finding a number. Plus or minus two. A differential equation is solved by finding a function. One whose derivatives satisfy the stated relationship. Consider the simplest meaningful example. Suppose a quantity y grows at a rate equal to itself — the more there is, the faster it grows. In symbols: d y d x equals y. A solution is any function y of x that makes that equation true when substituted in. The function y equals e to the x works, because its derivative is itself. But so does y equals five e to the x. And indeed y equals C e to the x, for any constant C. So already a hallmark of differential equations appears. They typically have not one solution, but an entire family of them.
Classification
Order, and Ordinary Versus Partial
Differential equations are classified two ways, and both matter for choosing a technique. First, by order. The order is the order of the highest derivative that appears in the equation. d y d x equals three x is first order — the highest derivative is the first. Second derivative of y, plus four y equals zero, is second order. And an equation involving a third derivative is third order. That single number largely determines the techniques required to solve the equation, and how many constants the solution family contains. Second, by whether they're ordinary or partial. An ordinary differential equation involves a function of a single independent variable, and therefore only ordinary derivatives. All of the equations in this chapter are ordinary. A partial differential equation involves a function of two or more independent variables, and uses partial derivatives. The heat equation and the wave equation — modeling temperature spreading through a metal plate, or vibrations traveling along a string — are the classic examples. Those are a substantial subject of their own, and we won't treat them here. Now here's a pattern worth noticing early, because it recurs throughout the chapter. The number of arbitrary constants in the complete solution family equals the order. A first-order equation yields a one-parameter family. A second-order equation, a two-parameter family. Which makes sense — each integration that undoes a derivative introduces exactly one constant.
Two Kinds of Solution
The Family, and One Member of It
Because a differential equation generally has a whole family of solutions, it's useful to distinguish between the family and its individual members. The general solution is the entire family, written with arbitrary constants. For d y d x equals two y, the general solution is y equals C e to the two x. Where C ranges over all real numbers. A particular solution is one specific function from that family, obtained by choosing a value for the constant. For instance y equals five e to the two x. That's the member with C equal to five. Now look at the picture, because geometrically this is very clean. The general solution is an infinite collection of curves, filling the plane. One curve for each value of C. Positive C values curve upward. Negative C values mirror below the axis. And C equals zero gives the flat line y equals zero. A particular solution selects a single curve from that collection. So the constants are exactly the freedom that extra information — an initial condition — will later remove.
Verification
Checking Is Easier Than Deriving
Before we solve anything, here's a skill worth having. You can confirm that a proposed function solves a differential equation without solving the equation from scratch. Simply differentiate the candidate, and substitute it in. If the equation reduces to a true statement for all x, the function is a solution. This verification skill is fundamental — because checking is often far easier than deriving. And it gives you a reliable way to catch errors. Let's verify that y equals three e to the two x is a solution of d y d x equals two y. Differentiate the candidate: three times two e to the two x. Which is six e to the two x. Now compute the right side: two y is two times three e to the two x. Also six e to the two x. The two sides agree. So it's a solution. Now a second-order one. Show that y equals C one cosine x plus C two sine x solves the second derivative of y, plus y, equals zero. Differentiate once: minus C one sine x, plus C two cosine x. Differentiate again: minus C one cosine x, minus C two sine x. Which is just minus y. Substitute into the left side: minus y plus y. Zero. Satisfied for all x, and for every choice of both constants. And note — this second-order equation carries two arbitrary constants. Matching its order, exactly as the pattern predicted.
Initial-Value Problems
Pinning Down the Constants
A general solution describes every possible behavior the system could have. But to predict the one behavior a real system actually exhibits, you need more information. Specifically, the state of the system at a starting moment. Such a specification is called an initial condition. And a differential equation packaged together with enough initial conditions to determine the constants is called an initial-value problem. For a first-order equation, an initial condition typically gives the value of the solution at one point. For a second-order equation, two conditions are needed — commonly the value and the first derivative at a point. Because two constants must be fixed. The procedure has three steps. Find the general solution. Substitute the initial conditions to set up equations for the constants. Then solve for them. Let's do a first-order one. Solve d y d x equals two y, with y at zero equal to five. The general solution is y equals C e to the two x. Apply the condition by substituting x equals zero and y equals five. Five equals C times e to the zero. Which is C times one. So C is five. The particular solution is y equals five e to the two x. That's the unique member of the family passing through the point zero comma five. Now a second-order one. Second derivative of y plus y equals zero, with y at zero equal to two, and y prime at zero equal to three. From the earlier example, the general solution is C one cosine x plus C two sine x. And its derivative is minus C one sine x plus C two cosine x. Apply the first condition at x equals zero. Cosine zero is one, sine zero is zero. So you get C one equals two. Apply the second condition to the derivative. You get C two equals three. Both constants determined. The particular solution is two cosine x, plus three sine x. Two conditions were required because the equation is second order. One condition would have left a constant undetermined. Here's something to think about. An initial condition doesn't have to be specified at x equals zero, despite the name. Suppose you'd been told y at one equals five, rather than y at zero. The general solution is unchanged. But C would come out differently. Why does the location of the condition matter for the value of C, but not for the form of the general solution? And what does that reveal about the distinct roles the differential equation and the initial condition each play?
Key Takeaways
One Pitfall, and What to Carry Forward
The most common conceptual error in this introduction is confusing the two kinds of solution. Reporting a general solution when a particular one was requested. Or the reverse. If a problem provides an initial condition, the expected answer is a single function with no remaining arbitrary constants. Leaving a C in the final answer means the initial condition was simply not used. Conversely, if no initial condition is given, the complete answer must retain its constants. Dropping them silently reports only one curve out of infinitely many, and loses information. A related mechanical slip is forgetting that a second-order problem needs two conditions. Supplying only one leaves the solution half-determined. The safeguard is simply to count. The number of arbitrary constants in the general solution should equal the order. And the number of initial conditions provided should match that same number, before a unique particular solution can be claimed. So — five takeaways. A differential equation relates a function to its derivatives, and its solution is a function, not a number. The order is the highest derivative present, and ordinary equations are distinguished from partial ones. The general solution is the full family with arbitrary constants. A particular solution fixes those constants to one curve. Verify any candidate solution by differentiating and substituting — checking is easier than deriving. And an initial-value problem supplies as many initial conditions as the order, determining the constants and selecting the unique particular solution. Next up is section five point two — Slope Fields and Euler's Method. For seeing and approximating solutions you cannot write down.
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