If `y = f (x+ct) + g(x-ct)`, then show that `(del^2y)/(delt^2)=c^2(del^2y)/(delx^2)`

If y=f(x+ct)+g(x-ct), then show that (del^(2)y)/(del t^(2))=c^(2)(del^(2)y)/(del x^(2))

If u= x^3-3xy^2 , show that (del^2u)/(delx^2)+(del^2u)/(dely^2) =0

If u = (x^2+y^2+z^2)^(-1/2) then prove that (del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2) =0

If u = x/y^2- y/x^2 , show that (del^2u)/(delxdely)=(del^2u)/(delydelx) .

If V(x,y) = e^(x) (x cos y - y siny) , then prove that (del^(2)V)/(del x^(2)) + (del^(2) V)/(del y^(2)) =0

Show that the equation, y=a sin(omegat-kx) satisfies the wave equation (del^(2) y)/(delt^(2))=v^(2) (del^(2)y)/(delx^(2)) . Find speed of wave and the direction in which it is travelling.

Show that the equation, y=a sin(omegat-kx) satisfies the wave equation (del^(2) y)/(delt^(2))=v^(2) (del^(2)y)/(delx^(2)) . Find speed of wave and the direction in which it is travelling.

Let w(x, y, z)=(1)/(sqrt(x^(2)+y^(2)+z^(2)))(x, y, z) != (0, 0, 0) . Show that (del^(2)w)/(delx^(2))+(del^(2)w)/(dely^(2))+(del^(2)w)/(delz^(2))=0

If w(x,y) = xy + sin(xy) , then prove that (del^(2) w)/(del y del x) = (del^(2) w)/(del x del y)

If z=f(x-cy) + F(x+cy) where f and F are any two functions and c is a constant, show that c^(2) (del^(2)z)/(del x^(2)) = (del^(2)z)/(del y^(2)) .