Relation between $f_{xx}$ and $f_{yy}$

Question

If $f(x,y) = \cos(cx+y) + e^{cx-y}$ Show $f_{xx}=c^{2}f_{yy}$ My try, $$f_{x}=-c\sin(cx+y)+ce^{cx-y}$$ $$f_{xx}=-c^{2}\cos(cx+y)+c^{2}e^{cx-y}$$ $$f_{y}=-\sin(cx+y)-e^{cx-y}$$ $$f_{yy}=-\cos(cx+y)+e^{cx-y}$$ So,$$ f_{xx}=c^2(-\cos(cx+y)+e^{cx-y})$$ so, $$f_{xx}=c^2f_{yy}$$ Is this approach correct? Is there another approach to show this?