4/1/11

This week I'm going to go over PARTICULAR SOLUTIONS. It's a really simple concept, but yet it's something that everyone forgets the steps for and cannot do. So to explain the concept...For the problem given to you, you're typically given a derivative already solved, in the form dy/dx = 'something'..And they'll tell you something along the lines of, "Find the particular solution (or they might just say "solution", but particular solution is implied) with initial condition f(#)=#..(which means they give you an x and y value that they want you to plug in after). Okay so here's a simple example explaining all of the steps thoroughly:

Ex. 1) Find the particular solution for 3y^2 + xy + 2x = 6 given that x=0 when y=1.
*Alright, so first you should notice the obvious--that you're not given what the derivative is. So they expect you to derive that equation before you start the particular solution
*So using implicit differentiation, you should get this as your first step when deriving the equation:
6ydy/dx + (x)(dy/dx) + (y)(1) + 2 = 0
which is
6ydy/dx + xdy/dx + y + 2 = 0
Now to simplify this you're going to have to get all the dy/dx's on one side and everything else on the other side:
6dy/dx + xdy/dx = -y-2
Now factor out a dy/dx
dy/dx(6+x) = -y-2
And divide by 6+x
dy/dx = (-y-2)/(6+x)
**Now we can use that to find our particular solution
*So the first step in finding the particular solution is to cross multiply..as in, multiply dy by 6+x and multiply dx by -y-2, and set them equal like this:
(6+x)dy = (-y-2)dx
*Alright now for the next step you're supposed to get the x's with the dx's and the y's with the dy's (by dividing usually)..butttttt, I don't see how that's possible with this equation..I mean I did make up this problem so that could be why, but this usually wouldn't happen on an AP. Soooo, for me to explain how to work the rest of this problem let's just pretend for our derivative we got this instead:
dy/dx = (6+x)/(-y-2)
*Now cross multiply and you should get this:
(-y-2)dy = (6+x)dx
*Now you're going to integrate both sides of the equation and put the "+C" on the right hand side. So after integrating you should have this:
-1/2y^2 - 2y = 6x + 1/2x^2 + C
*Now you're going to solve for C..And to do this you have to plug in the x and y value they gave you (x=0; y=1)
*So plugging in you should have:
-1/2(1)^2 - 2(1) = 6(0) + 1/2(0)^2 + C
C = 5/2
*Now plug your C into your integrated equation and solve for y to get your final answer
-1/2y^2 - 2y = 6x + 1/2x^2 + 5/2
*To solve this for y, you can first get rid of the fractions by multiplying every term by 2..After doing that you should have:
-y^2 - 4y = 12x + x^2 - 5
Rearranging that you have
-y^2 - 4y = x^2 + 12x - 5
Now you can factor out a -y..then you should get this:
-y(y+4) = x^2+12x-5
And divide by y+4
-y = (x^2+12x-5)/(y+4)
Divide by -1
y = - (x^2+12x-5)/(y+4) ..And that's your particular solution!