 So in general, an arbitrary differential equation can't be solved. And in a kind and gentle universe, you'd never be given a differential equation you couldn't solve. And fortunately, we don't live in that universe, and remember, your inability to solve a problem does not make the problem go away. And so the important question to ask is, what can we do with such equations? The general answer is going to be to apply some sort of numerical method. And we might go about it as follows. Suppose f of xy equals c defines a level curve, some sort of curve in the Cartesian plane. At any point, the derivative gives the slope of the line tangent to the curve at the point. And the tangent line is the best linear approximation to the curve near the point of tangency. And this suggests the following method for solving first-order differential equations. We'll start at our initial point, and we'll compute the derivative at that point. We'll follow the tangent line for a short distance to get to a new point. We'll compute the derivative at the new point, which gives us a new tangent line. We'll follow that for a short distance, and we'll lather, rinse, repeat. This method is known as Euler's method after an 18th century mathematician who specialized in, well, actually, everything. The short distance that we're following the tangent line, usually expressed in terms of the independent variable, is called the step size. For example, let's try to sketch a solution to this differential equation where we have an initial value. We'll use step size delta x equals 0.1, and then estimate y of 1. So the initial condition indicates the trajectory begins at the point 0e. At 0e, the derivative will be 0, which means the tangent to the trajectory at 0e has slope 0. Now, we'll follow the tangent line for delta x equal to 0.1, our step size. Remember, our slope is our change in y over the change in x, well, that's equal to 0. Delta x is 0.1, and so delta y is going to be 0. And if we put this together, we go from the point 0e, x increases by 0.1, y increases by 0, and so we arrive at the point 0.1e. Now, lather, rinse, repeat. At 0.1e, we have our derivative equal to, so the slope of the tangent line is 0.1, and with the step size of delta x equals 0.1, we find delta y is, and so x has increased by 0.1, y has increased by 0.1, and so we get to the point. Once more on to the breach, at this point, our derivative is, and so delta y will be, and so we'll move to the point. Now, you could do this by hand, but I wouldn't. You could program a computer to do all these steps automatically, but I'm a little lazy, so what I did is I did this in a spreadsheet format. So we have our initial values, x equals 0, y equals e. Our derivative, x log of y, delta y is going to be the derivative times our step size, 0.1. Our x value increases by the step size, 0.1. Our y value increases by the delta y value we just calculated. We'll compute our new derivative and new delta y value for the current point, and at the spreadsheet we can just fill down the formulas, and then we'll fill down all the formulas, and we can read off the value at x equal to 1. Now, that's with our step size delta x equals 0.1. If we decrease the step size, we hope we can increase the accuracy of our approximation. So if we use a step size of 0.01, and you really don't want to do this by hand, so use a computer spreadsheet, something like that, and we find that we eventually arrive at, which we hope gives us a better approximation to y of 1.