First off let me state this is homework…just not my homework (really…I’m 40 and fairly well done with school). I was asked the following and was hoping I could help. It is definitely beyond me though so thought I would ask here. I am led to believe the important parts are the steps as well as the solution. If because it is homework though and should not be answered here that’s cool…
1.) Solve explicitly, linear equation:
(x^2 + 1)dy/dx + 4xy = x, y(2)=1
2.) Solve explicitly the Bernouilli equation:
dy/dx + y = x*y^3
These are both solvable by the “integrating factors” technique. The idea is to notice that if you multiply the left-hand side by some fairly simple function, it becomes the derivative of a simple expression. For example, if you had the differential equation
x dy/dx + 2y = x
you might notice that (d/dx)(xy)=x dy/dx + y, which is not quite the left side, but (d/dx)(x[sup]2[/sup]y) = x[sup]2[/sup] dy/dx + 2xy = x(x dy/dx + 2y), which is just x times the left-hand side. So multiply the equation by x, and get
(d/dx)(x[sup]2[/sup]y) = x[sup]2[/sup] dy/dx + 2xy = x[sup]2[/sup] = (d/dx)(x[sup]3[/sup]/3)
and so the general solution is
x[sup]2[/sup]y = x[sup]3[/sup]/3 + C
for some constant C.
There are equations for the integrating factor (for example at the Wikipedia link above), but in many cases you can see what factor you need from the form of the equation. In your first equation you might be able to guess immediately at the required factor. In the second equation you need to manipulate the equation a little bit first to get it in the form of a linear equation.
Thanks Omphaloskeptic.
Frankly I am about to have an aneurysm sorting all that though. 