Stumped by a Dynamical Systems equation

[Vannicke]

Though the class is Dynamical Systems and Applications, which is hard enough on its own, I had a question on a test that simply has me stumped.

Solve dy/dx = 1/4 (x/4 + y)^4 .

The best I could get was setting u = x/4 + y so that u' = 1/4 + y' and getting:

1/4 - du/dx = 1/4 u^4

I recently covered Bernoullis Equation: y' + p(x) y = f(x) y^n but this doesn't fit into that, and I could not figure out any way to get it there.

The system doesn't seem to be seperable either, unless I missed something. On the test I had tried something with Laplace transforms, but I realize now that what I had done wasn't even close to the right way to use them.

I have tried for the past few hours to figure out something with:

du/(u^4 - 1) = -dx/4

getting:

integral{ 1/4 du/(u - 1) - 1/4 du/(u + 1) - 1/2 du/(u^2 + 1) } = x/4 + c

then:

1/4 ln(u-1) - 1/4 ln(u+1) - 1/2 arctan(u) = x/4 + c

then making a feeble attempt at solving that mess for u, which fried my brain.

Anyone want to give me a point in the right direction, or tackle this beast of an equation, or point and laugh cause i missed something obvious?


-Van

 

[tkhunny]

It looks to me like you have it. Where's the hangup?

Of course, you may want to start with "= dx" rather than "= dx/4" and simplify your life just a bit.

 

[Vannicke]

This was one of 8 problems in a test that i had only 50 minutes to complete ... and i have yet to actually find a solution for y, the arctangent i have been trying to wade through for hours, trying to find that rascal of a y.

Since it was a 50 minute test, I expected I was missing something here, something that should have been a ten minute solution, not the several hours i have put into this thing. I'm 99% certain what I've written here is what the test asked for, and I was one of the last people in the room, so either everyone else gave up on it, or there is something I've missed.

 

[duncan]

suggestion

you might consider:

(u^2+1)=(u+i)(u-i) where i^2 = -1

so: du/(u^2+1) = (du/(u+i)-du/(u-i)/(2i))

 

[galactus]

It would appear you were headed in the right direction.

\(\displaystyle \int\frac{1}{u^{4}+1}du=\frac{1}{4}\int dx\)

Do not worry about solving it for u. I doubt if that was expected, was it?.

This would be very difficult to solve for u.

If you managed to perform the integration and resub, I would think that should suffice.

I ran it through a DE solver and what it returned was not solved for x.

 

[duncan]

1/4 - du/dx = 1/4 u^4

Though the class is Dynamical Systems and Applications, which is hard enough on its own, I had a question on a test that simply has me stumped.

Solve dy/dx = 1/4 (x/4 + y)^4 .

The best I could get was setting u = x/4 + y so that u' = 1/4 + y' and getting:

1/4 - du/dx = 1/4 u^4 Should there be a minus sign?