If `(x+i y)^3=u+i v ,` then show that `u/x+v/y=4(x^2-y^2)`

If (x+iy)^(3)=u+iv then show that (u)/(x)+(v)/(y)=4(x^(2)-y^(2))

If u = x ^ (4), show that (y ^ (3) u) / (y x ^ (2)) = (^ (3) u) / (x and x)

If (x,y) and (x,y) are the coordinates of the same point referred to two sets of rectangular axes with the same origin and it ux+vx+vy , where u and v are independent of x and y becomes VX+UY, show that u^(2)+v^(2)=U^(2)+V^(2)

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v',v),(u'',v'',2v'):}| .

If (x, y) and (X, Y) be the coordinates of the same point referred to two sets of rectangular axes with the same origin and if ux + vy , when u and v are independent of X and Y become VX + UY, show that u^(2)+v^(2)=U^(2)+V^(2) .

If X and Y are subsets of the universal set U , then show that : (i) YsubXuuY (ii) XcapYsubX (iii) XsubYrArrXcapY=X .

Consider the differential equation. Dy/dx +P(x)y = Q(x) (i) If two particular solutions of given equation u(x) and v(x) are known, find the general solution of the same equation in terms of u(x) and v(x). (ii) If alpha and beta are constants such that the linear combinations alpha.u(x) + beta.v(x) is a solution of the given equation, find the relation between alpha and beta . (iii) If w(x) is the third particular solution different from u(x) and v(x) then find the ratio v(x) u(x)/w(x)-u(x)

Let a,b,c,d be real numbers in G.P. If u,v,w satisfy the system of equations u+2y+3w=6,4u+5v+6w=12 and 6u+9v=4 then show that the roots of the equation ((1)/(u)+(1)/(v)(+)/(w))x^(2)+[(b-c)^(2)+(c-a)^(2)+(d-b)^(2)]x+u+v+w=0 and 20x^^2+10(a-d)^^2x-9=0 ' are reciprocals of each other.

If u = log ((x ^ (2) + y ^ (2)) / (x + y)), prove that x (u) / (x) + y (u) / (y) = 1

If x and y are two uncorrelated variables such that u=x+y and v=x-y, find the coefficient of correlation betweeen u and v in terms of sigma_(x) and sigma_(y) .