From ca651ab00eb5ef4a5218ce262960cb87419b73da Mon Sep 17 00:00:00 2001 From: zacharias Date: Wed, 3 Dec 2025 14:53:28 +0100 Subject: [PATCH] vault backup: 2025-12-03 14:53:27 --- .obsidian/workspace.json | 43 +++++++++++++++++++++++++++------------- Definitioner.md | 4 +++- ODE.md | 23 +++++++++++++++++++++ 3 files changed, 55 insertions(+), 15 deletions(-) create mode 100644 ODE.md diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index b0c0443..261351b 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -49,6 +49,20 @@ "title": "Funktioner Forts" } }, + { + "id": "66f6bbd26d9be1ca", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "ODE.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "ODE" + } + }, { "id": "54baa3edd65a7c5d", "type": "leaf", @@ -69,12 +83,12 @@ "state": { "type": "markdown", "state": { - "file": "Gräsvärde (1).md", + "file": "Primära Funktioner.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Gräsvärde (1)" + "title": "Primära Funktioner" } }, { @@ -176,7 +190,7 @@ } } ], - "currentTab": 10 + "currentTab": 3 } ], "direction": "vertical" @@ -208,7 +222,7 @@ "state": { "type": "search", "state": { - "query": "\\sqrt", + "query": "\\text{p", "matchingCase": false, "explainSearch": false, "collapseAll": false, @@ -229,7 +243,8 @@ "title": "Bookmarks" } } - ] + ], + "currentTab": 1 } ], "direction": "horizontal", @@ -336,20 +351,21 @@ "obsidian-git:Open Git source control": false } }, - "active": "1b73858e17021be2", + "active": "66f6bbd26d9be1ca", "lastOpenFiles": [ - "Int1.png", + "Definitioner.md", + "ODE.md", "Primära Funktioner.md", - "Integraler.md", + "Derivata.md", "Differential.md", - "Tenta Example.md", + "Funktioner.md", + "Integraler.md", "Gräsvärde (1).md", - "Komplexa tal.md", "Grafer.md", "Funktioner Forts.md", - "Funktioner.md", - "Derivata.md", - "Definitioner.md", + "Int1.png", + "Tenta Example.md", + "Komplexa tal.md", "Def_graf1.png", "TE1.png", "Trigonometri.md", @@ -361,7 +377,6 @@ "gv1.png", "k2.png", "k1.png", - "f_inverse.png", "Untitled.canvas" ] } \ No newline at end of file diff --git a/Definitioner.md b/Definitioner.md index 1edbc0d..af19765 100644 --- a/Definitioner.md +++ b/Definitioner.md @@ -5,4 +5,6 @@ 2. *Värde på ändpunkten. (Eller gränsvärde)* 3. *Värde på punkter där derivata saknas(Kritiska punkter)* 4. *Jämför 1,2,3.* - - **Ex**: $$\begin{align}f(x)=1-\mid{x}\mid\\f'(0)\text{ Existerar inte}\end{align}$$![[Def_graf1.png]] \ No newline at end of file + - **Ex**: $$\begin{align}f(x)=1-\mid{x}\mid\\f'(0)\text{ Existerar inte}\end{align}$$![[Def_graf1.png]] +- **ODE**/**Primärfunktioner**/**Integraler** + - $$\begin{align}F'(x)=f(x)\\F(x)=\int f(x)dx\end{align}$$ \ No newline at end of file diff --git a/ODE.md b/ODE.md new file mode 100644 index 0000000..400ce21 --- /dev/null +++ b/ODE.md @@ -0,0 +1,23 @@ +**ODE** $\Longleftrightarrow$ **Ordinära differentialekvation** +**PDE** $\Longleftrightarrow$ **Partiell differentialekvation** +- Separabel ODE +- Linjär ODE av ordning 1 +- Linjär ODE av ordning 2 med konstant koefficienter + + +- **Ex**: *Newtons lag* $m\frac{d^2}{df^2}\stackrel{\rightarrow}{s}(t)=\stackrel{\rightarrow}{F}(t)$ +- **Ex**: **PDE** *Maxwellsekvation, Schrödingerekvation* $$\begin{align}\text{Okänd funktion }y(x)\\\text{ODE: }F\left(x,y(x),y'(x),\dots,y^{(n)}(x)\right)=0\\\text{Ording: }n\end{align}$$ +- **Ex**: *ODE av ordning 3*: $xy'''(x)+x^{1/4}y'(x)+\left(y(x)\right)^2=7x+3$ +- **Linjär ODE** + - $$\begin{align}a_n(x)y^{(x)}(x)+\dots+a_1(x)y'(x)+a_0(x)y(x)=h(x)\\a_k\text{ är funktionen av }x\end{align}$$ + - **Ex**: $$\begin{align}\sqrt{x}y''+\frac1xy'+\pi{y}=e^x\_\_(\star)\\\text{Om }y_{_1}\&y_{_2}\text{ uppfylen}\\\sqrt{x}y''+\frac1xy'+\pi{y}=0\_\_(1)\\\text{så är }\alpha{y_1}+\beta{y_2},\;\alpha,\beta\in\mathbb{R}\text{ också lösning till }(1)\end{align}$$ + - $(\star)$ är en *linjär ODE* + - **Ex**: $yy'=x+2$: *Icke-linjär* + - **Ex**: $\left.\begin{aligned}y'+\underline{\sqrt{y}}=x+2\\\underline{\sqrt{y'}}+y=2x+3\end{aligned}\right\}$: *Icke-linjär* + - **Ex**: $\underline{e^y}+\underline{\sin y}+y'=0$: *Icke-linjär* + - **Ex**: $(\sin x)y'+\sqrt{x}=\pi$: *Linjär* + - **Ex**: $$\begin{align}5y''=x+\sin x\\\Leftrightarrow y''=\frac15(x+\sin x)\\\text{Integrera m.a.p. }x\\y'=\frac15\int(x+\sin x)dx=\frac15\left(\frac{x^2}2-\cos x\right)+C\\y=\int\left(\frac15\left(\frac{x^2}2-\cos x\right)+C_1\right)dx\\=\frac15\left(\frac{x^3}6-\sin x\right)+C_1x+C_2\\\text{där }C_1,C_2\text{ är konstanter}\end{align}$$ +- $$\begin{align}\text{ODE: }g(y)y'=h(x)\\\text{Lösning: }g(y)y'=h(x)\\g(y)y'dx=h(x)dy\\\int g(x)dy=\int h(x)dx\\G(y)=H(x)+C\end{align}$$Där $G$ är primitiv till $g$ och $H$ är primitiv till $h$ +- **Ex**: $$\begin{align}y^2y'=x\sqrt{y}\;\;\left.\begin{aligned}\text{Icke-kin.}\\\text{ODE av}\\\text{ordning 1}\end{aligned}\right.\\\text{för }y\not\equiv0\\y^2y'=2x\sqrt{y}\\\Leftrightarrow\frac{y^2}{\sqrt{y}}y'=2x\Leftrightarrow y^{3/2}y'=2x\\\text{Integrera m.a.p. x}\\\int y^{3/2}y'dx=\int 2xdx\\\Leftrightarrow\int y^{3/2}dy=\cancel{2}\frac{x^{1+1}}{\cancel{1+1}}+C\\\Leftrightarrow y^{5/2}=\frac52\left(x^2+C\right)\\\Rightarrow y=\left[\frac52\left(x^2+C\right)\right]^{}2/5,C\in\mathbb{R}\\\text{Om }y(x)=0\;\forall{x}\in\mathbb{R},\text{ så är }y'(x)=0\\\left.\begin{aligned}\text{VL: }y^2y'?0^2\times0=0\\\text{HL: }2x\sqrt{y}=2x\times+=0\end{aligned}\right\}\;\;y(x)=0\text{ är en lösning}\\\underline{\text{Svar}}:y(x)=\left[\frac52\left(x^2+C\right)\right]^{2/5},\;x\in\mathbb{R}\\\text{eller }y(x)=0\end{align}$$ +- **Initialvärdersproblem** + - **Ex**: *Lös* **IVP** $$\begin{align}y^2y'=2x\sqrt{y},\;\;y(1)=1\\\underline{\text{Lösn}}:y=\left[\frac52\left(x^2+C\right)\right]^{2/5}\text{ eller }y=0\\y=0\text{ uppfyller inte vilkor }y(1)=1\\y(1)=1\\\Leftrightarrow\left[\frac52\left(1^2+C\right)\right]^{2/5}=1\\\Leftrightarrow\left[\frac52(10C)\right]^2=1^5=1\\\Leftrightarrow\frac52(1+C)=\pm1\\\Leftrightarrow1+C=\pm\frac25\Leftrightarrow\left\{\begin{aligned}-1+\frac25\\-1-\frac25\end{aligned}\right.\\\Leftrightarrow C=\frac{-3}5\text{ eller }\frac{-7}5\\\underline{\text{svar}}:y=\left[\frac52\left(x^2-\frac35\right)\right]^{2/5}\text{ eller}\\y=\left[\frac52\left(x^2-\frac75\right)\right]^{2/5}\end{align}$$ \ No newline at end of file