diff --git a/content/Homogenous equations (lec 2).md b/content/Homogenous equations (lec 2).md index 39b9c92..f9a2bb3 100644 --- a/content/Homogenous equations (lec 2).md +++ b/content/Homogenous equations (lec 2).md @@ -94,11 +94,11 @@ $A(x-y)=e^{y+2x}$ >$\lim_{ n \to 0 }\ln(n)=y+2x$ >$\lim_{ n \to 0 }\frac{d}{dx}\ln(n)=0=\frac{dy}{dx}+2$ >$\frac{dy}{dx}=-2\quad \Box$ ->so from $\frac{dy}{dx}=-2\frac{{x-y-\frac{1}{2}}}{x-y+1}$ we get: ->$-2=-2\frac{{x-y-\frac{1}{2}}}{x-y+1}$ ->$x-y+1=x-y-\frac{1}{2}$ ->$1=-\frac{1}{2}$ ->So what does this all mean? I honestly have no idea. I think it means we assumed that $e^{y+2x}=0$ is defined and because we arrived at a contradiction, our assumption was wrong. That didn't really get us to show if it was a valid solution or not like I imagined. +>plugging into the equation $(2x-2y-1)dx+(x-y+1)dy=0$ yields: +>$1=\frac{2(x-y+1)}{2x-2y-1}$ +>$2x-2y-1=2(x-y+1)$ +>$-1=2$ +>So what does this all mean? I think it means that even if we imagine that $\frac{dy}{dx}$ exists, the equation is not satisfied and $y=x$ is definitely not a solution even when we try to cheat a little. We can rearrange to our liking, but we have found the general solution to the DE: #### $$x-y=Ae^{2x+y}$$ diff --git a/content/Laplace transform (lec 14-16).md b/content/Laplace transform (lec 14-16).md index 7847585..ad52222 100644 --- a/content/Laplace transform (lec 14-16).md +++ b/content/Laplace transform (lec 14-16).md @@ -116,7 +116,7 @@ $\mathcal{L}^{-1}\{\alpha F(s)+\beta G(s)\}=\alpha \mathcal{L}^{-1}\{F\}+\beta \ This can be proven rather easily due to the linearity of the forward transform (wasn't done in class unfortunately). ## Examples -#ex #inv_LT I +#ex #inv_LT Compute this inverse LT: $$\mathcal{L}^{-1}\left\{ \frac{1}{s^5}+\frac{3}{(2s+5)^2}+\frac{1}{s^2+4s+8}+ \frac{{s+1}}{s^2+2s+10} \right\}$$ Notice that all these terms approach 0 as s approaches inf. diff --git a/content/Power series (lec 22-23).md b/content/Power series (lec 22-24).md similarity index 77% rename from content/Power series (lec 22-23).md rename to content/Power series (lec 22-24).md index b2700ae..69b4fc8 100644 --- a/content/Power series (lec 22-23).md +++ b/content/Power series (lec 22-24).md @@ -73,7 +73,7 @@ $\implies a_{n}=0$, $n=0,1,2,\dots$ #end of lec 22 #start of lec 23 Mid terms are almost done being marked! -## Examples +## Solving DE using series Let's start using power series to start solving DE! No magic formulas we need to memorize when solving equations using power series (Yay!) #ex @@ -153,4 +153,47 @@ $z(x)=a_{0}\left( 1+\sum_{k=1}^\infty \frac{(1*4*\dots(3k-2))^2}{(3k)!}x^{3k} \r $a_{1}\left( x+\sum_{k=1}^\infty \frac{(2\cdot 5\cdot \dots(3k-1))^2}{(3k+1)!} x^{3k+1}\right)$ there we go, $z$ is a linear combination of those two expressions class done at 1:56 (a lil late but the journey is worth it) -#end of lec 23 +#end of lec 23 #start of lec 24 +*midterms have been marked and returned today.* + +we consider: +$$y''+p(x)y'+q(x)y=0$$ +this is in standard form, it's a second order linear equation +Definition: +if $p(x)$ and $q(x)$ are **analytic** functions in a vicinity of $x_{0}$ then $x_0$ is **ordinary**. Otherwise, $x_{0}$ is **singular**. +we expect that the solution y can be represented by a power series. This is true according to the following theorem: +Theorem: If $x_{0}$ is ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$. +The radius of convergence for them is at least as large as the distance between $x_{0}$ and the closest singular point (which can be real or complex). +![[Drawing 2023-10-30 13.12.57.excalidraw.png]] + +## Examples for calculating $\rho$ +#ex +$$(x+1)y''-3xy'+2y=0 \quad x_{0}=1$$ +put it in standard form: +$y''-\frac{3xy'}{x+1}+\frac{2y}{x+1}=0$ +the only singular point for this equation is $x=-1$ +so the minimum value of radius convergence is $\rho=2$ (distance between -1 and x_0) +we are guaranteed that the power series will converge *at least* in $(-1,3)$, possibly more. You can try solving for y as a power series. + +#ex +$$y''-\tan xy'+y=0 \quad x_{0}=0$$ +notice the coefficient beside y is 1, 1 is analytic and differentiable everywhere, obviously! +Same goes for any polynomial, it's obvious that any polynomial is infinitely differentiable but it's important to know. +What about tan x? +$\tan x=\frac{\sin x}{\cos x}$ is not defined on $x=\frac{\pi}{2}\pm n\pi, \qquad n=0,1,2,\dots$ +the closest singular points are $\frac{\pi}{2}$ and $\frac{-\pi}{2}$ so our radius of convergence is the minimum distance of x_0 to these two points: +$\rho\geq\mid x_{0}-\frac{\pi}{2}\mid=\frac{\pi}{2}$ +convergence could be larger, but we are guaranteed convergence on $x=x_{0}-\rho$ to $x_{0}+\rho$ + +#ex +$$(x^2+1)y''+xy'+y=0 \qquad x_{0}=1$$ +put it in standard form: +$y''+\frac{x}{x^2+1}y'+\frac{y}{x^2+1}=0$ +remember singular points can be complex the two singular points are: +$x^2=1=0 \qquad x=\pm i$ +now we have to compute the two distances of these singular points to x=1 +![[Drawing 2023-11-03 13.40.54.excalidraw.png]] +To calculate distance: $\alpha_{1}+\beta_{1}i, \qquad \alpha_{2}+\beta_{2}i$ +$\sqrt{ (\alpha_{1}-\alpha_{2})^2+(\beta_{1}-\beta_{2})^2 }$ +$\rho\geq \sqrt{ 1^2+1^2 }=\sqrt{ 2 }$ +#end of lec 24 \ No newline at end of file diff --git a/content/Solving IVP's using Laplace transform (lec 17-18).md b/content/Solving IVP's using Laplace transform (lec 17-18).md index be72c33..1692477 100644 --- a/content/Solving IVP's using Laplace transform (lec 17-18).md +++ b/content/Solving IVP's using Laplace transform (lec 17-18).md @@ -36,7 +36,7 @@ $\frac{d^2y}{dx^2}-y=xe^x$ <- notice the y here is not the same as the y above, where $y(0)=0 \qquad \frac{dy}{dx}(0)=0$ Hit it with the LT! $\frac{1}{s^2}$ is LT of $x$. Using the shifting property, $\frac{1}{(s-\alpha)^2}$ is the LT of $xe^{\alpha x}$ -$s^2Y(s)-Y(s) =\frac{1}{(s-1)^2}$ +$s^2Y(s)-s\underbrace{ y(0) }_{ =0 }-\underbrace{ y'(0) }_{ =0 }-Y(s) =\frac{1}{(s-1)^2}$ Isolate $Y(s)$ : $(s^2-1)Y(s)=\frac{1}{(s-1)^2}$ $Y(s)=\frac{1}{(s-1)^2(s^2-1)}$ @@ -45,6 +45,16 @@ Partial fraction time: $Y(s)=\frac{1}{(s-1)^3(s+1)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)^3}+\frac{D}{s+1}$ $\frac{{A(s-1)^2(s+1)+B(s-1)(s+1)+C(s+1)+D(s-1)^3}}{(s-1)^3(s+1)}$ $\begin{matrix}A+D=0 \\A-2A+B-3D=0 \\ A-2A+B-B+C+3D=0 \\ A-B+C-D=1\end{matrix}$ + + + + + +$\begin{bmatrix}1 & 0 & 0 & 1 & 0 \\-1 & 1 & 0 & -3 & 0 \\-1 & 0 & 1 & 3 & 0 \\ 1 & -1 & 1 & -1 & 1\end{bmatrix}$ + +$-A+B-3D=0$ +$-A+C+3D=0$ + $A=\frac{1}{8}$ $B=-\frac{1}{4}$ $C=\frac{1}{2}$ diff --git a/content/_index.md b/content/_index.md index df94135..e596fdb 100644 --- a/content/_index.md +++ b/content/_index.md @@ -22,7 +22,7 @@ Good luck on midterms! <3 -Oct 18 2023 [Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.) [Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.) [Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html) (raw notes, not reviewed or revised yet.) -[Power series (lec 22-23)](power-series-lec-22-23.html) (raw notes, not reviewed or revised yet.) +[Power series (lec 22-24)](power-series-lec-22-24.html) (raw notes, not reviewed or revised yet.)
[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.) [Big LT table (.png)](drawings/bigLTtable.png) diff --git a/content/drawings/Drawing 2023-10-30 13.12.57.excalidraw.md b/content/drawings/Drawing 2023-10-30 13.12.57.excalidraw.md new file mode 100644 index 0000000..388d6f0 --- /dev/null +++ b/content/drawings/Drawing 2023-10-30 13.12.57.excalidraw.md @@ -0,0 +1,1034 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== + + +# Text Elements +%% +# Drawing +```compressed-json +N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATL + +ZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHAL + +RY8RMpWdx8Q1TdIEfARcZgRmBShcZQUARm0ATm1YhJo6IIR9BA4oZm4AbXAwUDBS + +iBJuCAARUgBJBNwAKwApKoB9dR59OB4AIQBWLLYjNNLIWERKgDNAhE8qfjLMbmce + +ADYebQBmdf6tgA4ABn3+hK2Adn7FyBgVnkOk/p4E/vPYvZ3Drdi+IsgKEjqbgAFj + +Wh2052BwNiwMOGyufykCEIymk3B4+206x4wP2sTW+wSwL2PwRYwg1mUwW4h2uEGY + +UFIbAA1ggAMJsfBsUiVADEsQQAoFozKmlw2GZyiZQg4xA5XJ5EkZ1mYcFwgRyIsg + 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13.12.57.excalidraw.png new file mode 100644 index 0000000..a665b43 Binary files /dev/null and b/content/drawings/Drawing 2023-10-30 13.12.57.excalidraw.png differ diff --git a/content/drawings/Drawing 2023-11-03 13.40.54.excalidraw.md b/content/drawings/Drawing 2023-11-03 13.40.54.excalidraw.md new file mode 100644 index 0000000..f3b715c --- /dev/null +++ b/content/drawings/Drawing 2023-11-03 13.40.54.excalidraw.md @@ -0,0 +1,828 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== + + +# Text Elements +%% +# Drawing +```compressed-json +N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATL + +ZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHAL + +RY8RMpWdx8Q1TdIEfARcZgRmBShcZQUARm0ATm1YhJo6IIR9BA4oZm4AbXAwUDBS + +iBJuCAAxAA5cbGdSAGsALQB1ABkEgFVWXHoKACkoAAkAdjTSyFhESoAzQIRPKn4y + +zG5nFIAGbS2eMdiANi3DgBZTgFYAZlieC9XIGA2eQ8PtA9ixhJSEnliasb3IqQCg + 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