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### Messages - Junhong Zhou

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##### Quiz 2 / Quiz2-6101 6D
« on: October 02, 2020, 02:22:25 PM »
Problem(3pt). Find all points of continuity of the given function;
$$f(z)= \begin{cases}\frac{z^4-1}{z-i},& z\neq i\\4i, & z=i \end{cases}$$

f(z) is continuous when $z\neq i$.

When z = i, then

$z^4-1=i^4-1=1-1=0$

$z - i = i-i = 0$

Now use the L'Hospital's Rule we have:
\begin{align*}
\lim_{z \to i} \frac{z^4-1}{z-i} &= \lim_{z \to i} \frac{4z^3}{1}\\
&= 4i^3\\
&= -4i\\
& \neq 4i
\end{align*}
Therefore f(z) is not continuous at z = i.

2
##### Quiz-3 / tut0402 quiz3
« on: October 11, 2019, 02:03:35 PM »
Question: Find the Wronskian of the given pair of functions.
$$cos^2(x), 1+cos(2x)$$

$$W = \det \begin{vmatrix} cos^2(x) & 1+cos2x \\ -2sin(x)cos(x) & -2sin(2x) \end{vmatrix} = \det \begin{vmatrix} cos^2(x) & 2cos^2(x) \\ -sin(2x) & -2sin(2x) \end{vmatrix}$$

\begin{align}
\implies W &= -2cos^2(x)sin2(x)-(-sin(2x))(2cos^2(x))\notag\\
&= 2sin(2x)(-cos^2(x)+cos^2(x))\notag\\
&= 2sin(2x)\cdot0\notag\\
&= 0\notag
\end{align}

3
##### Quiz-2 / TUT0402 quiz2
« on: October 04, 2019, 02:00:07 PM »
Question: Find an integrating factor and solve the given equation.
$$(3x^2y+2xy+y^3)+(x^2+y^2)y'=0$$

\begin{align}
M(x,y)=3x^2y+2xy+y^3 &\implies M_y=3x^2+2x+3y^2\notag\\
N(x,y)=x^2+y^2 &\implies N_x=2x\notag
\end{align}

Since $M_y \neq N_x$, this implies the given differential equation is not exact, so we need to find $\mu(x,y)$ such that the equation $\mu(3x^2+2xy+y^3)+\mu(x^2+y^2)y'=0$ is exact.

$$R(x)=\frac{M_y-N_x}{N}=\frac{3x^2+3y^2}{x^2+y^2}=3$$

then we can write $\mu$:
$$\mu(x,y)=e^{\int R(x)dx}=e^{\int 3 dx}=e^{3x}$$

multiply the given differential equation by $\mu$:
\begin{align}
\mu(3x^2y+2xy+y^3)+\mu(x^2+y^2)y' &= 0\notag\\
e^{3x}(3x^2y+2xy+y^3)+e^{3x}(x^2+y^2)y' &= 0\notag
\end{align}

Which is now an exact differential equation, this implies there exist $\varphi(x,y)$ such that $\varphi_x=M$ and $\varphi_y=N$.
\begin{align}
\varphi_y(x,y)=e^{3x}(x^2+y^2) &\implies \varphi(x,y)
=\int e^{3x}(x^2+y^2)dy\notag\\
&\implies \varphi(x,y)
= e^{3x}x^2y+\frac{1}{3}e^{3x}y^3+f(x)\notag
\end{align}
$$\varphi_x(x,y)=2e^{3x}xy+3e^{3x}x^2y+e^{3x}y^3+f'(x) \implies f'(x)=0 \implies f(x)=C\notag$$

Therefore:
$$\varphi(x,y)=e^{3x}x^2y+\frac{1}{3}e^{3x}y^3=C$$
$$e^{3x}x^2y+\frac{1}{3}e^{3x}y^3=C$$

4
##### Quiz-1 / TUT0402 quiz1
« on: September 27, 2019, 02:00:07 PM »
$$ty'+2y=sin(t)$$

write this into standard equation:
$$y'+\frac{2}{t}y=\frac{sin(t)}{t}$$

which here we have
$$p(t)=\frac{2}{t}$$
$$\mu(x)=e^{\int p(t)dt}=e^{\int\frac{2}{t}dt}=e^{2\ln{t}}=t^2$$

multiply $\mu(x)$ to both sides of the standard equation we have:
\begin{align}
(t^2)y'+(t^2)\frac{2}{t}y &= (t^2)\frac{sin(t)}{t}\notag\\
t^2y'+2ty &= tsin(t)\notag\\
\int t^2y'+2tydt &=\int tsin(t)dt\notag\\
t^2y &= \int tsin(t) \notag
\end{align}

integrate $\int tsin(t)dt$ by parts, let $u=t$ and $dv=sin(t)dt$:
\begin{align}
\int tsin(t)dt &= t(-cos(t))-\int -cos(t)dt\notag\\
&= -tcos(t)+\int cos(t)dt\notag\\
&= -tcos(t)+sin(t)+C\notag
\end{align}

so we have:
\begin{align}
t^2y &= -tcos(t)+sin(t)+C\notag\\
y &= \frac{-tcos(t)+sin(t)+C}{t^2}\notag
\end{align}

So:
$$\lim_{t\to\infty}y=\lim_{t\to\infty}\frac{-tcos(t)+sin(t)+C}{t^2}=0$$

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