4.3.1

Instructor: Example 1-- find the general solution to the differential equation y prime equals quantity x squared minus 4 multiplied by quantity 3y plus 2, using the method of separation of variables. I'm leaving the problem solving strategy for separation of variables here for just a moment for you to look over. But we're just going to go straight and apply this. 

So the first thing we are going to do is rewrite this differential equation. And I'm going to rewrite it as dy/dx-- rather than y prime-- equals x squared minus 4, 3y plus 2 multiplied. Now I'm going to rewrite this. If you can think of multiplying by dx on both sides and then dividing by 3 by plus 2, but I'm going to rewrite this as 3y plus 2 dy equals x squared minus 4 dx. 

Oh, something's not right there. We're dividing by 3, not plus 2. Now actually what this form is written in that problem-solving strategy. So you can check that out. 

Now once we have this, we want to take the antiderivative on the left side with respect to y and on the right side with respect to x. And that would be-- and I'm going to spare you some of the integration technical details and just say this is 1/3 natural log the absolute value of 3y plus 2. And x squared would become x cubed over 3 minus B, minus, 4x plus C. And I'm going to go and just C1 for the time being. You'll see why I do that. 

All right, well we want to isolate y, if at all possible. And so first thing we're going to do is multiply both sides through by 3. I'm going to go ahead and do this. That will produce an equation that is natural log of absolute value of 3y plus 2z equals x cubed minus 12x plus-- and I'm going to say I'm really going to have a 3C1. Let's just call that our constant number two. We can often sympathize toward the very end with just a C. So I'm going to keep numbering these until I get to my final solution. 

Now to isolate the y value, I want to raise or exponentiate-- raise both sides with a base of e exponentiation. That will make this left side 3y plus 2. And we're going to drop-- it turns out actually, I can drop the absolute values as long as I allow for my constant in just a moment to be positive or negative. Now that would be-- I would have e to the x cubed minus 12x. And my exponent or my constant there is actually going to become a coefficient. 

So this is going to be e to the C2. And let's just say that is C3. And make that C3. Now subtracting 2 from both sides and then dividing by 3, I can obtain-- and now, I have to say C3. I'm going to drop the absolute value bars, because I'm going to just allow for it to be positive or negative. That comes up several times. 

So that will be C3 e to the x cubed minus 12x minus 2, all divided by 3. And at this point, this is the point where I would want to say, OK, well, let's just drop the subscript, just call it C. And that is our general solution.

[ENDS AT 4:22]