Consider the differential equation e^(x+y) dy/dx = e^(x-y). What is (d²y/dx²)(dx/dy)² equal to?

From e^(x+y) dy/dx = e^(x-y), we get e^y · dy/dx = e^(-y-y)·... actually rewriting: dy/dx = e^(x-y)/e^(x+y) = e^(-2y). So e^(2y) dy = dx. Differentiating dy/dx = e^(-2y) with respect to x: d²y/dx² = -2e^(-2y)·dy/dx = -2e^(-2y)·e^(-2y) = -2e^(-4y). Also dx/dy = e^(2y). So (d²y/dx²)(dx/dy)² = -2e^(-4y)·e^(4y) = -2.