I'm doing some studying for a test and I have a question about differential equations. So we have this one exercise which says: $u_x=u_y$ and $u(x,y)=X(x)*Y(y)$
I do what I've got to do and find that $X(x) = Ae^{lx}$ and $Y(y) = Be^{ly}$ so $u(x,y) = ABe^{l(x+y)}$. The next exercise says that $u_{xx} = ut$ so I do $X''(x)/X(x) = T'(t)/T(t)$. I solve this equation and I get $X(x) = c_1e^{\sqrt{l}*x} + c2e^{-\sqrt{l}x}$ and $T(t) = Ae^{lt}$ but in the book $T(t) = e^{lt}$. So why doesn't it have a constant in front of $e^{lt}$? I find to by solving this equation: $T'(t) - l*T(t) = 0$.