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I have an the equation $2U_t + U_{xt} = 0$. I found the general solution as $$U(x,t)=e^{-2x}P(t)+Q(x)$$ I have two questions about this:

1-find the solution for $U(x,0)=0$; $U_t(x,0)=e^{-2x}$

2-Is there a solution for $U(x,0)=0$; $U_t(x,0)=1$

I will appreciate any suggestion.

EditPiAf
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Shay
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1 Answers1

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For your first question, imposing the general solution in your boundary condition gives $$ e^{-2x}P(0)+Q(x)=0\\e^{-2x}P'(0)=e^{-2x} $$ then any $P, Q$ satisfying $ P'(0)=1, Q(x)=-P(0)e^{-2x} $ gives a solution.

For the second question, I don't think there exist a solution. If imposing your general solution to the second boundary condition, it gives $ P'(0)=e^{2x} $, which means that $P$ has to be a function of $x$. That's a contradiction.

Gao
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  • My main problem is: how can P′(0) be used in this question? it is not used anywhere. so what can this condition be used for? – Shay Jan 22 '19 at 20:27
  • For example, for the first question, $P'(0)$ has to equal 1, which means function $P(x)$ in solution can not be arbitrary. – Gao Jan 23 '19 at 23:23