The most widely solvable type of non-linear equation is called a separable equation. These include autonomous equations that we were just talking about. More generally, it's anytime x prime can be written down as a function of x times a function of t. If we use the Leibnitz notation on the derivative, it makes it more memorable. The title kind of gives away the plot. We separate the variables, all the x's go with the left, all the t's go on the right. Then you integrate both sides. And if you can do the integrals, got a solution. Here's an example. We have dx, d t is to t x divide both sides by x. And now we can integrate both sides. On the left, we get natural log of absolute value of x. On the right we have t squared plus an arbitrary constant. I'll exponentiate both sides. I'll rewrite e to the c as a new constant a by it has to be positive because it's the exponential of a real constantly. To take off the absolute value signs I have to use plus or minus. That's very curious that we're leaving out just one case for a, which is when a is 0, what about that one? If you go back and look at the original problem, 0 is a steady-state solution. So we can write out that the solution is a times e to the t squared with no restrictions on a. Why did this weird extra case happen? Basically, it's because we divided by x, that rules out any possibility that x is 0. So you have to check that separately. In this example, it's not clear that we have a separable equation until we factor on the right hand side. Now we can write the d x, d t is x times 1 minus t over t. So we put the x on the left and t's on the right. Notice that we've divided by T and we divided by x. So we're going to make a note of that. Integrate both sides. We'll write e to the c as a positive constant. But taking away the absolute values gives us a plus or minus times a positive constant a. So really any non-zero, hey, will work. The reason we had to rule out 0 was because we divided by x. If we try that as a constant steady solution in the original equation, we see that it works. So we can write the solution is constant times t minus t with no restrictions. On a. Note that this problem is linear. So we could have solved it that way as well. In this example, we have an initial value problem. We put x is on the left and t's on the right. Integrate. When we do the integrations, we don't need to put a constant on both sides. We confuse them into a single constant of integration. And you stare at this and you realize you haven't solved a lot of quartic equations in your life. It's super hard. So it's best to just leave the solution in this form. This is an implicit form for the solution. It relates x and t, but without giving you an explicit formula for x as a function of t. However, we still can find this arbitrary constant c by plugging in the initial condition. We put in t equals 0 and x equals 4 simultaneously. Then we find the value of C.
VII.3 Separable equations
From Tobin Driscoll April 22, 2021
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