SOLUTION: This question relates to finding partial derivatives. Burger's Equation is a partial differential equation, used for describing wave processes in acoustics and hydrodynamics, (

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Question 798503: This question relates to finding partial derivatives.
Burger's Equation is a partial differential equation, used for describing wave processes in acoustics and hydrodynamics,
(let d= partial derivative, a=alpha, and L=lambda)
+%28dw%2Fdt%29=+%28%28d%5E2w%2Fdx%5E2%29%2B+w%28dw%2Fdx%29%29%29
verify that
w%28x%2Ct%29=+L+%2B+%282%2F%28x%2BLt%2B+a%29%29
is a solution, where L and a are arbitary constants
thank-you for any help
Found 2 solutions by KMST, rothauserc:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
That's a lot to write, so I will define
Y%28x%2Ct%29=1%2F%28x%2BLt%2B+a%29=%28x%2BLt%2B+a%29%5E%28-1%29

Now w%28x%2Ct%29=+L+%2B+2Y is much easier to write.

dw%2Fdt=+%28dw%2FdY%29%28dY%2Fdt%29=+2%28-1%28x%2BLt%2B+a%29%5E%28-2%29%2AL%29=-2LY%5E2





That was the calculus part.
The rest is just algebra:

Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
We are going to take two partial derivatives of the equation
w(x, t) = L + (2 / (x +Lt +a)
rewite the equation as
w(x, t) = L + 2*(x +Lt +a)-1
note that L and a are arbitrary constants
@w/@t = 2*-1*L*(x +Lt +a)-2 = -2L/(x +Lt +a)^2
note that for the above partial derivative x is constant
@w/@x = 2*-1*(x +Lt +a)^-2 = -2 / (x +Lt +a)^2
note that for the above partial derivative t is constant
@w/@x ----> @^2w/@x^2 = @w/@x (-2 * (x +Lt +a)^-2 = 4 / (x +Lt +a)^3
now we can calculate
@^2w/@x^2 + w*@w/@X
= 4 / (x +Lt +a)^3 + (L +2 / (x +Lt +a)) * (-2 / (x +Lt +a)^2)
= 4 / (x +Lt +a)^3 + -2L / (x +Lt +a)^2 -4 / (x +Lt +a)^3
= -2L / (x +Lt +a)^2
= @w/@t, as required.