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**Chapter 2 / Re: Week 2 Lec 1 (Chapter 2) question**

« **on:**January 17, 2022, 07:48:38 PM »

Now it is correct $x=Ce^{t}$ and then $C=?$

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Now it is correct $x=Ce^{t}$ and then $C=?$

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Display formulae are surrounded by double dollars and no empty lines. Multiline formulae use special environments (google LaTeX gather align

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1. Do not use $*$ as a multiplication sign!

2. Do not use LaTeX for italic text (use markdown of the forum--button I)

3. Escape ln, cos, .... : \ln (x) to produce $\ln (x)$ and so on

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Yes, all linear are also semilinear and all semilinear are also quasilinear. For full mark you need to provide the most precise classification. So, if equation is linear you say "linear", if it is semilinear but not linear you say "semilinear but not linear" and so on,... "quasilinear but not semilinear" and "non-linear and not quasilinear".

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In particular, the definition of a linear PDE, from the textbook, is: $au_{x}+bu_{y}+cu-f=0$, where $f= f(x,y)$. However, if we simply move the the $cu$ to the right-hand side, we get: $au_{x}+bu_{y}=f-cu$. Now, define $g(x,y,u) = f(x,y)-cu$, then $au_{x}+bu_{y}=g(x,y,u)$, and the right-hand side now depends on lower-order derivatives, so by definition, it's quasi-linear. Could someone help identify the issue with this argument?First, it will be not just quasilinear, but also semilinear. Second, it will also be linear since you can move $c(x,y)u$ to the left

Good job, you mastered some $\LaTeX$ basics.

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We replace differentiation by $x$, y$ by multiplication on $\xi,\eta$. So $\partial_x^2 \mapsto \xi^2$ (just square); as a result senior terms like $Au_{xx}+2Bu_{xy}+ Cu_{yy}$ are replaced by quadratic form $A\xi^2+2B\xi\eta+C\eta^2$.

In the Linear Algebra you studied quadratic forms, right? And you know that

In the Linear Algebra you studied quadratic forms, right? And you know that

- if $AC-B^2 >0$ the canonical form is $\pm (\xi^2+\eta^2)$ (as $\pm A>0$)
- if $AC-B^2 <0$ the canonical form is $ (\xi^2-\eta^2)$,
- if $AC-B^2 =0$, but at least one of coefficients is not $0$ the canonical form is $\pm \xi^2$.

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Solution is allowed to be discontinuous.

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You need to indicate that there are no zeroes on $\gamma$

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"How ellipses wouls look like" means the directions and relative size of their semi-axis. See frame 4 of MAT244_W8L3 handout

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There could be misprints

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For higher order equations it is covered in MAT244-LEC0201-W6L2 (see modules). It is mandatory material.

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you need to write it, if you hope for any answer

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Indeed, instead of $\log(\pm w)$ with $+2\pi mi$ we write $\log(w)$ with $+\pi mi$ since $\log (-w)=\log(w)+\i i$

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Of **some** fundamental set (remember a constant factor!)

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Yes, some of them are geometric series, and some of $e^{z}$, $\sin(z)$, $\sinh(z)$ and so on. However some can be derived from those, ether by substitution (f.e. $z^2$ instead of $z$), some by integration, differentiation, multiplication by $z^m$ or combination of both. F.e. consider geometric $\dfrac{1}{1-z}$. Integratinfg we can get power series for $-\Log (1-z)$, diffeerentiating for $\frac{1}{(1-z)^m}$ ,...