Question

How do I solve the differential equation x^4(dy/dx)-xy^2=-3x5^5?

Answer 1

this is a 1st order nonlinear ordinary differential equation and its solution is quite involved. You will get a family of solutions unless you have an initial condition, which will give you one trajectory. A more qualitative way is to plot dy/dx as a function of x and y to find fixed points, saddles, maximums, minimums, etc. then you can extrapolate to y as a function of x for a particular trajectory (i.e. certain initial condition). See http://www.wolframalpha.com/input/?i=solve++x^4%28dy%2Fdx%29-xy^2%3D-3x5^5

Answer 2

I have four options for the answer and I am not sure if I treat the equation as seperable or if i apply the integrating factor rule. The answer bank show that when you solve for y, I will get either y=x^2, y=-x^2, y=3z, or y=3x^2.

Answer 3

So you want the analytical solution to:
\[x^4y' - xy^2 = -3x^5 \ \ ?\]

Answer 4

Yeah, I can't seem to find what y(x) would be. My choices are y=x^2, y=-x^2, y=3x, and y=3x^2. When I try to move the x's and y's to either side of the equal sign to begin integrating, I don't get the right answer.

Answer 5

Use a trial function (sometimes called Ansatz) of y = Ax^n.

Substitute this function into the equation and you end up with

\[An x^{n+3} - Ax^{2n+1} = -3x^5\]

This equality can only hold if there are equal powers of x on each side, so n + 3 = 5 and 2n + 1 = 5. Namely n = 2.

Simplify\[2A x^5 - A x^5 = Ax^5 = -3 x^5. \ \ \hbox{Hence} \ \ A = -3\]

Thus the solution is y(x) = Ax^n = 3x^5.

Make sense?

Answer 6

Ugh. Sorry, I mean Ax^n = -3x^2 !!

Answer 7

Ha yeah that does make sense. I t just seems simpler than it my professors taught me. Can I only apply that to differentials in that form?

Answer 8

More or less, yes. ODEs a tricky cats: some can be skinned more than one way and this I think is the best way to deal with this one.

If you like my answer, please give me a 'good answer' ;-)