If x and y be real, show that the equation : `sin^2 theta= (x^2+y^2)/(2xy)` is possible only when `x =y ne 0`.

If x and y are real, show that the equation : sec^2 theta= (4xy)/(x+y)^2 is valid only when x = y ne 0 .

Show that the equation sec^2 theta=(4xy)/(x+y)^2 is only possible when x=y

If sin^(2)theta=(x^(2)+y^(2)+1)/(2x) . Find the value of x and y.

Solve the differential equations: x^2y'=x^2+xy+y^2

Solve the differential equations: y^2dx+(x^2-xy+y^2)dy=0

If x,y in R and satisfy the equation xy(x^(2)-y^(2))=x^(2)+y^(2) where xne0 then the minimum possible value of x^(2)+y^(2) is

Solve the differential equation : (x^2y + x^2)dx + (xy^2 - y^2) dy = 0

The number of all possible values of theta , where 0 lt theta lt pi , for which the system of equations (y+z) cos 3 theta = (xyz) sin 3 theta x sin 3 theta=(2 cos 3 theta)/y+(2 sin 3 theta)/z (xyz) sin 3 theta = (y+2z) cos 3 theta +y sin 3 theta has a solution (x_(0), y_(0), z_(0)) with y_(0)z_(0) ne 0 is _______.

Show that the function ,y, defined by y(x) = sqrt(1+x^(2)) (x in R) is the solution of (1+x^(2))y' = xy