few fixes and added temporary message on index page

content/Heat equation (lec 30-33).md

@@ -194,7 +194,7 @@ $a_{0}=\frac{2}{\pi}\int _{0} ^{\pi/2} 1\, dx=1$
 $a_{n}=\frac{2}{\pi}\int _{0} ^{\pi/2} 1\cdot\cos(nx) \, dx=\frac{2}{n\pi}\sin(nx)|_{0} ^\frac{\pi}{2}=\frac{2}{n\pi}\sin\left( \frac{n\pi}{2} \right)  \quad n=1,2,\dots$
 We can take out the zero terms:
 $a_{2k}=0 \qquad a_{2k-1}=\frac{2}{(2k-1)\pi}(-1)^{k+1}$
-$u(t,x)=\frac{1}{2}+\frac{2}{\pi}\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}e^{-(2k-1)^2t}\cos((2k-1)x)$
+$$u(t,x)=\frac{1}{2}+\frac{2}{\pi}\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}e^{-(2k-1)^2t}\cos((2k-1)x)$$
 We are done, and we didn't need to take out the zero terms but if you want to be diligent, then there you go.
 Plot (only showing 100 harmonics, that's why the red line looks a lil' wiggly):
 ![plot](drawings/insulatedheat.png)

content/Periodic functions (lec 19).md

@@ -43,7 +43,7 @@ $$\mathcal{L}\{f\}=\frac{1}{s(1+e^{-as})}$$
 
 #ex #IVP #periodic #second_order_nonhomogenous #LT #partial_fractions
 Solve for $y(t)$ in the following second order periodic equation:
-$$y''+3y'+2y=f(t) \qquad y(0)=y'(0)=0 \quad a=1$$where $f(t)$ is from the previous example and $a$ is the width of $\frac{1}{2}$ of a period in the function $f(t)$
+$$y''+3y'+2y=f(t) \qquad y(0)=y'(0)=0 \quad a=1$$where $f(t)$ is from the previous example and $a$ is the width of $\frac{1}{2}$ of a period in the function $f(t)$. This means $f(t)$ has a period of $T=2$
 Hit it with the LT!
 $s^2Y+3sY+2Y=\mathcal{L}\{f\}= \frac{1}{s(1+e^{-1s})}$
 $(s+1)(s+2)Y= \frac{1}{s(1+e^{-1s})}$

content/Wave equation (lec 33-36).md

@@ -55,11 +55,11 @@ btw this equation models the electromagnetic field, to some approximation.
 the lowest mode is called the fundamental mode, the following terms after are called harmonics.
 If two instruments play the same note (same fundamental frequency), they sound different! and that's because of their difference in harmonics.
 The modes are standing waves in the string.
-"my claim, and this is not just my claim [...], is that any object, including social objects, can be described by waves. [...] Everything is a wave."
+"My claim, and this is not just my claim [...], is that any object, including social objects, can be described by waves. [...] Everything is a wave."
 You can model elementary particle behaviours with solitons (non linear waves.)
 In life in the real world, all waves have finite speed.
 So that's why its important to learn the wave equation. It's the prototype to waves.
-"waves are the fundamental object. [...]. So that's why it's important, these are the fundamental objects of nature here."
+"Waves are the fundamental object. [...]. So that's why it's important, these are the fundamental objects of nature here."
 </br>
 So, let's try an example using our formula above:
 $f(x)=\begin{cases}x, & 0\leq x\leq \frac{\pi}{2}\\ \ \pi-x, & \frac{\pi}{2}<x\leq \pi\end{cases}$

content/_index.md

@@ -2,6 +2,9 @@
 These are notes for the [University of Alberta MATH 201 - Differential Equations](https://apps.ualberta.ca/catalogue/course/math/201) course.
 I have written these notes for myself, I thought it would be cool to share them. These notes may be inaccurate, incomplete, or incoherent. No warranty is expressed or implied. Reader assumes all risk and liabilities. 
 </br>
+Good luck on the final! <3
+If we do bad on the exam, Petar will come after us with the Dirac delta 🤜💥
+</br>
 [Separable equations (lec 1)](separable-equations-lec-1.html)
 [Homogenous equations (lec 2)](homogenous-equations-lec-2.html)
 [Linear equations (lec 2-3)](linear-equations-lec-2-3.html)