If u =(x^2 + y^2)/(x + y), find the constant k such that x dy/dx + y du/dy = k
To find the constant k such that x(dy/dx) + y(du/dy) = k, we can use the given function u = (x^2 + y^2)/(x + y) and differentiate it partially with respect to x and y.
1. Finding dy/dx:
To find dy/dx, we can first express u as a function of x and y:
u = (x^2 + y^2)/(x + y)
Now, we can implicitly differentiate u with respect to x using the quotient rule:
du/dx = [(x + y)(2x) - (x^2 + y^2)(1)] / (x + y)^2
Simplifying further:
du/dx = [2x^2 + 2xy - x^2 - y^2] / (x + y)^2
du/dx = [x^2 + 2xy - y^2] / (x + y)^2
Since we are given x(dy/dx) + y(du/dy) = k, we need to find dy/dx.
2. Finding du/dy:
To find du/dy, we differentiate u with respect to y:
du/dy = [(x + y)(2y) - (x^2 + y^2)(1)] / (x + y)^2
Simplifying further:
du/dy = [2xy + 2y^2 - x^2 - y^2] / (x + y)^2
du/dy = [2xy + y^2 - x^2] / (x + y)^2
Again, since we are given x(dy/dx) + y(du/dy) = k, we need to solve for dy/dx.
3. Solving for dy/dx:
We can rearrange the given equation to solve for dy/dx:
x(dy/dx) + y(du/dy) = k
Substitute the expressions we found for du/dx and du/dy:
x(dy/dx) + y(du/dy) = x(dy/dx) + y([2xy + y^2 - x^2] / (x + y)^2) = k
Multiply through by (x + y)^2 to clear the denominator:
x(x + y)(dy/dx) + y(2xy + y^2 - x^2) = k(x + y)^2
Simplify:
x^2(dy/dx) + xy(dy/dx) + 2xy^2 + y^3 - xy^2 - yx^2 = k(x^2 + 2xy + y^2)
Combine like terms:
x^2(dy/dx) + xy(dy/dx) + xy^2 + y^3 = k(x^2 + 2xy + y^2)
Now, use the fact that (x^2 + y^2)/(x + y) = u:
(xy^2 + y^3)/(x + y) = k(x^2 + 2xy + y^2)
Simplify further:
y(x^2 + y^2) = k(x^2 + 2xy + y^2)(x + y)
y(x^2 + y^2) = k(x^3 + 2x^2y + xy^2 + x^2y + 2xy^2 + y^3)
y(x^2 + y^2) = k(x^3 + 3x^2y + 3xy^2 + y^3)
Now, we can equate the coefficients of each term on both sides of the equation to find the value of k.
For x^3 term:
0 = kx^3
For x^2y term:
0 = 3kx^2y
For xy^2 term:
y(x^2 + y^2) = 3kxy^2
For y^3 term:
0 = ky^3
From these equations, we can observe that the only possible value for k such that all the equations hold true is k = 0.
Therefore, the constant k in x(dy/dx) + y(du/dy) = k is k = 0.