 So let's start by solving a couple of simple differential equations. Now before we do that, we need to identify what it is to solve a differential equation. So if we want to solve a differential equation in standard form, and again standard form as we have solved for the highest order derivative in terms of the other derivatives and the independent variable, we want to find a function f of t that solves the equation. And the good news is you've already solved some differential equations. And that's because you've used the fundamental theorem of calculus before. And again, remember there are several different versions of the fundamental theorem, but one of them is the following. Suppose f prime of t is equal to some function. Then capital F of t is one of the antiderivatives of f of t. So let's try to solve f prime of t is equal to 5. So our fundamental theorem of calculus says that if we know the derivative, then the function is going to be one of the antiderivatives. And so we want to find an antiderivative of 5. So the antiderivative of 5 is 5t. But don't forget, any time we find an antiderivative, there's always a constant of anti-differentiation. So in fact our antiderivative is 5t plus c. And so our function is 5t plus c. The constant of anti-differentiation means that there isn't a single solution to the differential equation. Instead there's a family of solutions which differ by the constant. We might also say that f of t equals 5t plus c is the general solution. And again, how you speak influences how you think. When we talk about a family of solutions, this is a more geometric concept of the solution and we might think about our solutions as curves. On the other hand, when we talk about a general solution, this is a more algebraic concept of the solution and we might think about our solutions as functions. And an important idea to keep in mind for now is that the solution to a differential equation will have at least one undetermined constant of anti-differentiation. So let's try to solve y prime of t is equal to y of t. And so we might proceed as follows. We want to find an anti-derivative of y of t and so we want y of t to be the anti-derivative of y of t. Unfortunately, this has y of t on both sides so it's not entirely clear how we can solve this. Instead, we have to remember an important derivative. The derivative of e to the x is e to the x. And so we might proceed as follows. Let y of t equal e to the t. Oh, wait a minute. We're supposed to have at least one undetermined constant of anti-differentiation. So let's make that plus c. That means y prime of t is going to be e to the t. And, well, I do need to solve this equation. In order for y prime of t to be equal to y of t, we have to have e to the t plus c has to be equal to e to the t. And that means that c has to be 0. And so our solution is y of t equals e to the power of t. But wait a minute. The solution to a differential equation will have at least one undetermined constant of anti-differentiation. We're supposed to have an undetermined constant here and we don't. So how do we get one? So one of the things we might remember is that when you differentiate constant times a function, that's going to be the same as constant times the derivative of the function. So instead of letting y of t be e to the t plus some constant, we'll let y of t be some constant times e to the t plus some other constant. And to distinguish the two constants, we'll call this one c1 and this one c2. Now you might be a little bit concerned here that we're throwing these constants down and what's it going to cost us? And here's a useful thing to remember. Trust the algebra. In practice what this means is you can include as many additional constants as you want, the algebra will take care of the ones you don't need. So if my function is c1 e to t plus c2, the derivative will be, and since I want y to be y prime of t, equals means replaceable, so I'll replace. And the algebra tells me c2 has to be zero, but c1 is still undetermined. And so our solution is y of t equals c1 e to the power t, where we do have that undetermined constant. And now let's think about this. If we want to find a solution to a differential equation, we don't have to worry about c other than making sure that it's actually there. But differential equations are most useful when they apply to specific situations, so we need to find the solution to a differential equation, and this requires we find the values of the undetermined constants. And to solve for c, we need the initial value, or more generally, the boundary values. When we found values for the undetermined coefficients that meet all of the initial and or boundary values, we have found the particular solution to the differential equation. So in our equation f prime of t equals 5, we might know that f of 0 is equal to 8. So we found that f of t equals 5t plus c is a family of solutions, and so we want the cousin where f of 0 is equal to 8. So we have our family, and we know something about f of 0. So let's see what our equation says about f of 0. E equals means replaceable, so f of 0 is equal to 8, and so that tells me that c is equal to 8. And so we can say that the particular solution is f of t equals 5t plus 8. Or again, we want to solve y prime of t equals y of t, where y of 0 is equal to 3. And so we found our family of solutions, y of t equals c e to power t, and we can solve for c. And so we know something about y of 0, and that tells us that c is equal to 3, and so the solution is y of t equals 3e to power t.