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### Messages - Ziyuan Wang

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##### Web Bonus Problems / Re: Web bonus problem -- Week 2
« on: January 15, 2018, 02:38:01 PM »
Differentiating Ioana's $(A)$ and equating it with his(her?) $(B)$, we have the symbolic system
$$\left( \begin{array}{cc|c} 1 & -1 & 2x\\ 1+x& 1-x&3x^{2}\\ \end{array} \right) \implies \left( \begin{array}{c} \varphi'(X)\\ \psi'(Y) \end{array} \right) = \left( \begin{array}{c} X\\ -Y \end{array} \right) \text{ where X, Y are the arguments of \varphi, \psi resp. } \implies \left( \begin{array}{c} \varphi(X)\\ \psi(Y)\end{array} \right) = \left( \begin{array}{c} X^{2}/2+C_{1}\\ -Y^{2}/2+C_{2} \end{array} \right) \implies u=(x+t)^{2}/2 - (x-t)^{2}/2 + Cx+D.$$

The last two terms in the final step are empirical, and I urgently seek a theoretical account for it.
u=(x+t)2/2-(x-t)2/2, where u is define on t>=0 and x is any real number.

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##### Web Bonus Problems / Re: Web bonus problem -- Week 2
« on: January 14, 2018, 06:28:20 PM »
Hint: What is the general solution of (\ref{A})?

For (1) it is $$u(t,x)= \phi(x+t) + \psi(x-t)$$

For the Cauchy Problem it is:

$$u(t,x)=\frac{1}{2}\bigl[g(x+t)+g(x-t)\bigr]+ \frac{1}{2}\int_{x-t}^{x+t} h(y)\,dy$$

I thought sol'n to Cauchy problem only works in IVP with t=0.

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##### Web Bonus Problems / Re: Web bonus problem -- Week 2
« on: January 14, 2018, 05:00:30 PM »
Is it u(x,t)=C1(x+t)+C2(x-t). Then I plug in t=x2/2 for u and ut and I get C1(x+x2/2)= (x3+x2)/2 and C2(x-x2/2) = (x3-x2)/2 which is the question i asked before. Can you tell me a bit more? I still don't get it.

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##### Web Bonus Problems / Re: Web bonus problem -- Week 2
« on: January 14, 2018, 01:56:43 PM »
May I ask how do we make a function of x+x2/2 equals (x3+x2)/2. I have been thinking for a long time and still can't figure it out.

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