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removed index message and added a few fixes

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content/Free vibrations (lec 11-12).md
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@ -83,7 +83,7 @@ $\phi \approx 0.9273\dots$
$$y(t)=\frac{5}{4}e^{-3t}\sin(4t+\phi)$$ $$y(t)=\frac{5}{4}e^{-3t}\sin(4t+\phi)$$
Important take away: We computed $\phi$ and $A$ in this example. We found a way to know the envelope of the amplitude of the oscillating system and it's phase shift. Important take away: We computed $\phi$ and $A$ in this example. We found a way to know the envelope of the amplitude of the oscillating system and it's phase shift.
"I know engineers love calculators, I know mathematicians hate calculators, and that's probably the only difference between mathematicians and engineers." -Prof (referring to calculating arctan(4/3) on an exam) <i>"I know engineers love calculators, I know mathematicians hate calculators, and that's probably the only difference between mathematicians and engineers."</i> -Prof (referring to a student question on calculating arctan(4/3) on an exam. Btw the answer is no, you wouldn't need to evaluate that on an exam.)
3.) b=10 3.) b=10
$r_{1,2}=-\frac{10}{2}\pm \frac{\sqrt{ 10^2-4*25 }}{2}=-5$ (repeated root, critically damped) $r_{1,2}=-\frac{10}{2}\pm \frac{\sqrt{ 10^2-4*25 }}{2}=-5$ (repeated root, critically damped)
$y(t)=(c_{1}+c_{2}t)e^{-5t}$ $y(t)=(c_{1}+c_{2}t)e^{-5t}$

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content/Wave equation (lec 33-36).md
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@ -95,7 +95,7 @@ $$u(0,t)=u(\pi,t)=0 \qquad t>0$$
$$u(x,0)=\sin(x) \qquad 0\leq x\leq \pi$$ $$u(x,0)=\sin(x) \qquad 0\leq x\leq \pi$$
$$\frac{ \partial u }{ \partial t }(x,0)=5\sin(2x)-3\sin(5x)\qquad 0\leq x\leq \pi$$ $$\frac{ \partial u }{ \partial t }(x,0)=5\sin(2x)-3\sin(5x)\qquad 0\leq x\leq \pi$$
when $tx$ wasn't there in last problem we had the solution: when $tx$ wasn't there in last problem we had the solution:
$u(t,x)=\sum_{n=1}^\infty \underbrace{ (a_{n}\cos(nt)+b_{n}\sin(nt)) }_{ u_{n}(t) }\sin(nt)$ notice $L=\pi$ $u(t,x)=\sum_{n=1}^\infty \underbrace{ (a_{n}\cos(nt)+b_{n}\sin(nt)) }_{ u_{n}(t) }\sin(nx)$ notice $L=\pi$
since $u(0,t)=u(\pi,t)=0$ we can expect a solution of the form: since $u(0,t)=u(\pi,t)=0$ we can expect a solution of the form:
$u(x,t)=\sum_{n=1}^\infty u_{n}(t)\sin\left( \frac{n\pi x}{\pi} \right)$ $u(x,t)=\sum_{n=1}^\infty u_{n}(t)\sin\left( \frac{n\pi x}{\pi} \right)$
If any of the boundary conditions are non zero, then we have to split(?) into X and T. (needs verification) If any of the boundary conditions are non zero, then we have to split(?) into X and T. (needs verification)
@ -142,9 +142,10 @@ Here's a plot showing the behavior of the string (graphed up to 40 harmonics):
Finished the solution. Man I got teary eyed from this lecture. Finished the solution. Man I got teary eyed from this lecture.
#end of lec 35 #end of lec 35
#start of lec 36 #start of lec 36
# Last lecture
What do you guys wanna do? Questions or summary of the course? What do you guys wanna do? Questions or summary of the course?
Okay we do summary. Okay we do summary.
# Summary of second half of Math 201 ## Summary of second half of Math 201
(available on eclass) (available on eclass)
Laplace transforms: Laplace transforms:
Definition of laplace, Definition of laplace,

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content/_index.md
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@ -2,9 +2,6 @@
These are notes for the [University of Alberta MATH 201 - Differential Equations](https://apps.ualberta.ca/catalogue/course/math/201) course. 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. 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> </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) [Separable equations (lec 1)](separable-equations-lec-1.html)
[Homogenous equations (lec 2)](homogenous-equations-lec-2.html) [Homogenous equations (lec 2)](homogenous-equations-lec-2.html)
[Linear equations (lec 2-3)](linear-equations-lec-2-3.html) [Linear equations (lec 2-3)](linear-equations-lec-2-3.html)
@ -30,9 +27,14 @@ If we do bad on the exam, Petar will come after us with the Dirac delta 🤜💥
[Fourier series (lec 28-29)](fourier-series-lec-28-29.html) [Fourier series (lec 28-29)](fourier-series-lec-28-29.html)
[Heat equation (lec 30-33)](heat-equation-lec-30-33.html) [Heat equation (lec 30-33)](heat-equation-lec-30-33.html)
[Wave equation (lec 33-36)](wave-equation-lec-33-36.html) [Wave equation (lec 33-36)](wave-equation-lec-33-36.html)
</br> </br>
[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.) [How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st 2023. Needs revision, but it gives a nice overview.)
[Big LT table (.png)](drawings/bigLTtable.png) [Big LT table (.png)](drawings/bigLTtable.png)
[Small LT table (.png)](drawings/smallLTtable.png) [Small LT table (.png)](drawings/smallLTtable.png)
</br> </br>
# Recommended material:
For the midterm exam, I highly recommend watching this video by The Math Sorcerer: [youtube.com/watch?v=kIZpbeE_yTc](https://youtube.com/watch?v=kIZpbeE_yTc)
From my experience, studying off this video was by far the best use of my time. Try each question yourself and follow his solution to check.
</br>
For the final exam, I unfortunately couldn't find good study videos. I recommend studying PDE's hard, solidify your understanding of heat eq, driven heat eq, heat eq with non-zero end points, wave eq, and driven wave eq. Afterwards, I recommend studying power series since it's the next biggest scary monster. Finally, go over the rest of the past topics to fill your understanding and memory if you have the time. I'm thinking I should record a final exam review guide, hmmm. I'll update this text if I ever make one.
</br>