Question

if Z= eˆx (x cos y - y sin y) show that ∂²z/∂x² + ∂²z/∂y² = 0

Answer 1

Use ln on both sides of the equation to remove the 'e'
Now use the derivative implicitly
find dx
and likewise dy
I hope it helped
good studies

Answer 2

∂z/∂x = e^x (xcosy - ysiny) + e^xcosy
∂²z/∂x²=e^x(xcosy -ysiny) + e^x cosy +e^x cosy
=xe^x cosy -ye^x siny + 2e^x cosy

∂z/∂y=e^x(-xsiny -siny -ycosy)
∂²z/∂y²= e^x (-xcosy -cosy -cosy +ysiny)
= -xe^x cosy -2e^x cosy +ye^x siny

∂²z/∂x² +∂²z/∂y² = (xe^x cosy -ye^x siny + 2e^x cosy) + (-xe^x cosy -2e^x cosy +ye^x siny) = 0 (proved)