Find a, b given a function satisfies the Laplace equation. The function is $f(x,y,z)=e^{ax+by}\cos(5z)$. I am supposed to find a and b given that it satisfies Laplace equation: $f_{xx}+f_{yy}+f_{zz}=0$
So, I found each derivative and replaced it back into the equation:
$\cos(5z)e^{ax+by}(a^2+b^2-25)=0$
So, all I can say is that $a^2+b^2=25$ which doesn't solve the problem. Am I missing something? 

$f(x,y,z)=e^{ax+by}\cos(5z)$.
let us find derivatives  seperately
1.$f_{x}=a*e^{ax+by}\cos(5z)$
2.$f_{xx}=a^2*e^{ax+by}\cos(5z)$
3.$f_{yy}=b^2*e^{ax+by}\cos(5z)$
4.$f_{z}=-5*e^{ax+by}\sin(5z)$
5.$f_{zz}=-25*e^{ax+by}\cos(5z)$
not  if we arrange similar terms we get  the same as you got it
$\cos(5z)e^{ax+by}(a^2+b^2-25)=0$
clearly solution is  all $a,b$ which satisfy this solution  and they  are
$(-3,-4)$,
$(-3,4)$,
$(-4,-3)$,
$(-4,3)$,
$(3,4)$,
$(4,3)$,
$(-5,0)$,
$(0,-5)$,
$(0,5)$,
$(5,0)$
that is all.good lucks
