fixed a mistake in seperation of var&eigen value problems andmade other misc changes

content/Second order homogenous linear equations (lec 5-7).md

@@ -4,14 +4,15 @@ most of these "models" in EE are based on these DE. You'll see how important DE
 Second order equations arise from very simple problems many engineers face, for instance a pendulum can be described by a second order equation.
 #second_order
 ### $$a_{2}(t)y''+a_{1}(t)y'+a_{0}(t)y=f(t)$$
-To motivate our interest: #fix
+To motivate our interest:
 ![draw](drawings/Drawing-2023-09-15-13.32.48.excalidraw.png)
-$ma=my''=-by'-ky$ 
+$F=ma=my''$
+$my''=-by'-ky$ 
 Look how a second order equation describes the motion of a mass-spring system!
 > Circuits that contains resistors, capacitors and inductors also behaves with this equation as well if you ignore the external magnetic fields around the circuit.
 
-The equation $my''+by'+ky=0$ is a homogenous second order equation. (in this case, it's full name is homogenous second order linear equation with constant coefficients.) 
+The equation $my''+by'+ky=0$ is a homogenous second order equation, because the RHS is 0. (in this case, it's full name is homogenous second order linear equation with constant coefficients.) 
->Similar pattern with the electrical circuit analogy. This DE ignores external forces on the mass-spring system, it only considers the friction and the spring. If we push the mass then there would be an external force.
+>Similar pattern with the electrical circuit analogy. This DE ignores external forces on the mass-spring system, it only considers the friction and the spring. If we push the mass then there would be an external force and the RHS would be non zero, and the equation would be non homogenous.
 
 It's called second order because we have second derivative in the equation.
 

content/Separation of variables & Eigen value problems (lec 26-28).md

@@ -122,7 +122,7 @@ this is a separable equation.
 We can treat the function $T$ as a variable:
 $\frac{dT}{dt} \frac{1}{T}=-\left( \frac{n\pi}{L} \right)^2D$
 $\int{dT} \frac{1}{T}=\int-\left( \frac{n\pi}{L} \right)^2Ddt$
-$\ln(T)=-\left( \frac{n\pi}{L} \right)Dt+c_{n}$
+$\ln\mid T \mid=-\left( \frac{n\pi}{L} \right)^2Dt+c_{n}$
 $T_{n}(t)=c_{n}e^{-(\frac{n\pi}{L})^2Dt}$
 >Yes this looks illegal, but it works, you could also integrate more rigorously if you did a u-sub: $u=T(t) \quad \frac{du}{dt}=T'(t)$)
 </br>

content/_index.md

@@ -32,14 +32,14 @@ I have written these notes for myself, I thought it would be cool to share them.
 [Big LT table (.png)](drawings/bigLTtable.png)
 [Small LT table (.png)](drawings/smallLTtable.png)
 </br>
-# Recommended study material
+# Additional recommended study 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. 
+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>
 The recommended course textbook when I took the class was: <i>Fundamentals of Differential Equations, R. Kent Nagle, Edward B. Saff and Arthur D. Snider, 9th Edition</i> Which is a good textbook imo, although I didn't use it often.
 </br>
-I mostly studied the material by attending the lectures and then reviewing/revising these typed notes on the bus or at home, often relying on my prof's notes on eclass in case I copied off the whiteboard wrong/couldn't keep up. (eclass is the name of my university's online class management system.)
+Personally, I studied the material by attending the lectures and then reviewing/revising these typed notes at home, often relying on my prof's notes on eclass in case I copied off the whiteboard wrong/couldn't keep up. (eclass is the name of my university's online class management system.)
-Of course there may still be mistakes riddled throughout so as of Jan 5th 2024, <b>I'm offering 1$ CAD in bounties for every mistake reported to my email/git repo, at least until supplies last.</b> General editing and formatting changes are also gladly welcomed through the git repository below or by email. Seeing people use these notes and benefitting from it makes me happy, so thanks for sticking around :) and remember to use what you learn for good! And to lead life with honor and integrity and be ethical engineers. Dr. Minev used to never forget to stress the importance of this in his lectures, and I wholeheartedly agree.
+Of course there may still be mistakes riddled throughout so as of Jan 5th 2024, <b>I'm offering 1$ CAD in bounties for every mistake reported to my email/git repo, at least until supplies last.</b> General editing and formatting changes are also gladly welcomed through the git repository below or by email.
 </br>

content/drawings/Drawing-2023-09-15-13.32.48.excalidraw.md

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 ```
 %%

themes/zettels/layouts/index.html

@@ -2,6 +2,7 @@
 	<p style="text-align: center;"><a href="http://sasserisop.com">Back to Sasserisop homepage</a></p>
 	{{ partial "content.html" . }}
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+<p style="text-align: left;">Seeing people use these notes and benefitting from it makes me happy, so thanks for sticking around :) and remember to use what you learn for good! And to lead life with honor and integrity and be ethical engineers. Dr. Minev used to always stress the importance of this in his lectures, and I wholeheartedly agree.</p></br>
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