# Problem with differential equation

I'm doing some studying for a test and I have a question about differential equations. So we have this one exercise which says: $ux = uy$ 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: $T'(t) - l*T(t) = 0$.

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Because X(x) already contains a multiplicative constant (multiply c1 and c2 by a given parameter) so it is not useful to plug another one in T(t) since only the product X(x)T(t) is meaningful. Thus, to write T(t)=c3e^(lt) is not false, just useless. – Did Aug 27 '11 at 13:58
I've added LaTeX formatting to your question, captain - apologies if I changed your intended meaning (it was tough to tell whether you wanted $\sqrt{l}\cdot x$ or $\sqrt{lx}$, for example). – Zev Chonoles Aug 27 '11 at 14:06
What do $ux=uy$ and $u_{xx}=ut$ mean? – Christian Blatter Aug 27 '11 at 17:44