Initial Value Problems

Consider the following situation. You’re given the function f(x)=2x-3 and asked to find its derivative. This function is pretty basic, so unless you’re taking calculus out of order, it shouldn’t cause you too much stress to figure out that the derivative of f(x) is 2. Not too awful, right?

Now consider what it would be like to work backwards from our derivative. If we’re given the function f’(x)=2 and asked to find its integral, it’s impossible for us to get back to our original function, f(x)=2x-3. As you can see, taking the integral of the derivative we just found gives us back the first term of the original function, 2x, but somehwere along the way we lost the -3. In fact, we always lose the constant (term without a variable attached), when we take the derivative of soemthing. Which means we’re never going to get the constant back when we try to integrate our derivative. It’s lost forever.

Accounting for that lost constant is why we always add C to the end of our integrals. C is called the “constant of integration” and it acts as a placeholder for our missing constant. In order to get back to our original function, and find our long-lost frined, -3, we’ll need some additional information about this problem, namely, an initial condition, which looks like this:

Problems that provide you with one or more initial conditions are called Initial Value Problems. Initial conditions take what would otherwise be an entire rainbow of possible solutions, and whittles them down to one specific solution.

Let’s take another look at this example in the box to the right.

Remember that the basic idea behind Initial Value Problems is that, once you differentiate a function, you lose sopme information about that function. More specifically, you lose the constant. By integrating the derived function, you get a family of solutjions that all have the same derivatives as f(x) and only differ by the unknown and myseterious constant.

Given one point on the function, (the initial condition), you can pick a specific solution out of a much broader solution set.