vault backup: 2025-12-03 14:53:27
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43
.obsidian/workspace.json
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43
.obsidian/workspace.json
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@@ -49,6 +49,20 @@
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"title": "Funktioner Forts"
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"title": "ODE"
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@@ -69,12 +83,12 @@
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"title": "Gräsvärde (1)"
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"title": "Primära Funktioner"
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@@ -361,7 +377,6 @@
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@@ -6,3 +6,5 @@
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3. *Värde på punkter där derivata saknas(Kritiska punkter)*
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4. *Jämför 1,2,3.*
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- **Ex**: $$\begin{align}f(x)=1-\mid{x}\mid\\f'(0)\text{ Existerar inte}\end{align}$$![[Def_graf1.png]]
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- **ODE**/**Primärfunktioner**/**Integraler**
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- $$\begin{align}F'(x)=f(x)\\F(x)=\int f(x)dx\end{align}$$
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23
ODE.md
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23
ODE.md
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**ODE** $\Longleftrightarrow$ **Ordinära differentialekvation**
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**PDE** $\Longleftrightarrow$ **Partiell differentialekvation**
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- Separabel ODE
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- Linjär ODE av ordning 1
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- Linjär ODE av ordning 2 med konstant koefficienter
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- **Ex**: *Newtons lag* $m\frac{d^2}{df^2}\stackrel{\rightarrow}{s}(t)=\stackrel{\rightarrow}{F}(t)$
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- **Ex**: **PDE** *Maxwellsekvation, Schrödingerekvation* $$\begin{align}\text{Okänd funktion }y(x)\\\text{ODE: }F\left(x,y(x),y'(x),\dots,y^{(n)}(x)\right)=0\\\text{Ording: }n\end{align}$$
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- **Ex**: *ODE av ordning 3*: $xy'''(x)+x^{1/4}y'(x)+\left(y(x)\right)^2=7x+3$
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- **Linjär ODE**
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- $$\begin{align}a_n(x)y^{(x)}(x)+\dots+a_1(x)y'(x)+a_0(x)y(x)=h(x)\\a_k\text{ är funktionen av }x\end{align}$$
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- **Ex**: $$\begin{align}\sqrt{x}y''+\frac1xy'+\pi{y}=e^x\_\_(\star)\\\text{Om }y_{_1}\&y_{_2}\text{ uppfylen}\\\sqrt{x}y''+\frac1xy'+\pi{y}=0\_\_(1)\\\text{så är }\alpha{y_1}+\beta{y_2},\;\alpha,\beta\in\mathbb{R}\text{ också lösning till }(1)\end{align}$$
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- $(\star)$ är en *linjär ODE*
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- **Ex**: $yy'=x+2$: *Icke-linjär*
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- **Ex**: $\left.\begin{aligned}y'+\underline{\sqrt{y}}=x+2\\\underline{\sqrt{y'}}+y=2x+3\end{aligned}\right\}$: *Icke-linjär*
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- **Ex**: $\underline{e^y}+\underline{\sin y}+y'=0$: *Icke-linjär*
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- **Ex**: $(\sin x)y'+\sqrt{x}=\pi$: *Linjär*
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- **Ex**: $$\begin{align}5y''=x+\sin x\\\Leftrightarrow y''=\frac15(x+\sin x)\\\text{Integrera m.a.p. }x\\y'=\frac15\int(x+\sin x)dx=\frac15\left(\frac{x^2}2-\cos x\right)+C\\y=\int\left(\frac15\left(\frac{x^2}2-\cos x\right)+C_1\right)dx\\=\frac15\left(\frac{x^3}6-\sin x\right)+C_1x+C_2\\\text{där }C_1,C_2\text{ är konstanter}\end{align}$$
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- $$\begin{align}\text{ODE: }g(y)y'=h(x)\\\text{Lösning: }g(y)y'=h(x)\\g(y)y'dx=h(x)dy\\\int g(x)dy=\int h(x)dx\\G(y)=H(x)+C\end{align}$$Där $G$ är primitiv till $g$ och $H$ är primitiv till $h$
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- **Ex**: $$\begin{align}y^2y'=x\sqrt{y}\;\;\left.\begin{aligned}\text{Icke-kin.}\\\text{ODE av}\\\text{ordning 1}\end{aligned}\right.\\\text{för }y\not\equiv0\\y^2y'=2x\sqrt{y}\\\Leftrightarrow\frac{y^2}{\sqrt{y}}y'=2x\Leftrightarrow y^{3/2}y'=2x\\\text{Integrera m.a.p. x}\\\int y^{3/2}y'dx=\int 2xdx\\\Leftrightarrow\int y^{3/2}dy=\cancel{2}\frac{x^{1+1}}{\cancel{1+1}}+C\\\Leftrightarrow y^{5/2}=\frac52\left(x^2+C\right)\\\Rightarrow y=\left[\frac52\left(x^2+C\right)\right]^{}2/5,C\in\mathbb{R}\\\text{Om }y(x)=0\;\forall{x}\in\mathbb{R},\text{ så är }y'(x)=0\\\left.\begin{aligned}\text{VL: }y^2y'?0^2\times0=0\\\text{HL: }2x\sqrt{y}=2x\times+=0\end{aligned}\right\}\;\;y(x)=0\text{ är en lösning}\\\underline{\text{Svar}}:y(x)=\left[\frac52\left(x^2+C\right)\right]^{2/5},\;x\in\mathbb{R}\\\text{eller }y(x)=0\end{align}$$
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- **Initialvärdersproblem**
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- **Ex**: *Lös* **IVP** $$\begin{align}y^2y'=2x\sqrt{y},\;\;y(1)=1\\\underline{\text{Lösn}}:y=\left[\frac52\left(x^2+C\right)\right]^{2/5}\text{ eller }y=0\\y=0\text{ uppfyller inte vilkor }y(1)=1\\y(1)=1\\\Leftrightarrow\left[\frac52\left(1^2+C\right)\right]^{2/5}=1\\\Leftrightarrow\left[\frac52(10C)\right]^2=1^5=1\\\Leftrightarrow\frac52(1+C)=\pm1\\\Leftrightarrow1+C=\pm\frac25\Leftrightarrow\left\{\begin{aligned}-1+\frac25\\-1-\frac25\end{aligned}\right.\\\Leftrightarrow C=\frac{-3}5\text{ eller }\frac{-7}5\\\underline{\text{svar}}:y=\left[\frac52\left(x^2-\frac35\right)\right]^{2/5}\text{ eller}\\y=\left[\frac52\left(x^2-\frac75\right)\right]^{2/5}\end{align}$$
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