4.1.3

Instructor: As we've seen with antiderivatives, we can find general solutions and particular solutions. Consider a differential equation y prime equals 2x. Well, the general solution to that, the function that whose derivative is 2x is x squared plus, C, some constant. So there are many possible solutions to this for varying values of C. These are all graph of different functions. 

So let's particularly find the particular solution to the differential equation y prime equals 2x passing through the point 2 comma 7. So if it's written in a general form, then with a plus C, then that is a general solution, but the particular solution will actually be a function and one exact function. So we have a whole family of them. And we're going to find one of them that works. 

So to find a solution to this, we could-- I mean, we could find the antiderivative. And we know that we have y equals x squared plus C. That's our general family of functions whose derivative is 2x. 

Now we want to take that 0.27 as an xy coordinate. We're going to say this is 7 equals 2 squared plus C1. And in general, you want parentheses there. Well, 2 squared 4 so that C is 3. So our particular solution is y equals x squared plus.