`xy = log y + C : y' = (y^(2))/(1 - xy)(xy ne 1)`

y = x sin x : xy' = y + x sqrt(x^(2) - y^(2))(x ne 0 and x gt y or x lt -y)

y = Ax : xy' = y (x ne 0)

y = sqrt(1 + x^(2)) : y' = (xy)/(1 + x^(2))

Verify (i) x ^(3) + y ^(3) = (x + y) (x ^(2) -xy + y ^(2)) (ii) x ^(3) -y ^(3) = (x-y) (x ^(2) +xy +y ^(2)) using some non-zone positive integers and check by actual multiplication. Can you call these as identities ?

(x^(2) - y^(2)) dx + 2xy dy = 0

If f(x) + f(y) = f((x+y)/(1-xy)) for all x, y in R (xy ne 1) and lim_(x rarr 0) (f(x))/(x) = 2 , then

Show that the general solution of the differential equation (dy)/(dx) + (y^(2) + y + 1)/(x^(2) + x + 1) = 0 is given by ( x + y + 1) = A(1 - x - y - 2xy) , where A is parameter.

** is a binary operation o Z. If x"*" y = x^(2)+y^(2)+xy then find [(1"*"2)+(0"*"3)]^(2) .

A = x + y and B = xy x = (20 pm1) cm , y = (10 pm 1) cm Then which of the following is correct.

|a| lt 1 and |b| lt 1, x=1 + a + a^(2) + ……. and y=1 + b + b^(2) + …..,Prove that, 1 + ab + a^(2) b^(2) + ….= (xy)/(x+y-1)