vault backup: 2025-12-09 14:41:50

.obsidian/workspace.json

@@ -243,8 +243,7 @@
               "title": "Bookmarks"
             }
           }
-        ],
-        "currentTab": 1
+        ]
       }
     ],
     "direction": "horizontal",
@@ -353,15 +352,15 @@
   },
   "active": "66f6bbd26d9be1ca",
   "lastOpenFiles": [
-    "Definitioner.md",
-    "ODE.md",
-    "Primära Funktioner.md",
-    "Derivata.md",
-    "Differential.md",
     "Funktioner.md",
-    "Integraler.md",
     "Gräsvärde (1).md",
+    "Primära Funktioner.md",
+    "Differential.md",
+    "Integraler.md",
     "Grafer.md",
+    "ODE.md",
+    "Definitioner.md",
+    "Derivata.md",
     "Funktioner Forts.md",
     "Int1.png",
     "Tenta Example.md",

ODE.md

@@ -33,6 +33,12 @@
 	- *Eftersom **ODE** är linjär, superpositionsprincip ger att* $\left.\begin{aligned}y_h\;\;\text{homohen lösning}\\y_p\;\;\text{Partikulär lösning}\end{aligned}\right\}\Longrightarrow y_h+y_p\;\;\text{också en lösning.}$
 	- **Karakteristiska polynomet**: $p(r)=r^2+ar+b$
 	- **Karakteristiska ekvationen**: $p(r)=0$
+	- **Homogena lösningar**: *Fall 1: Karakteristiska polynomet har reella rötter $r_1$ och $r_2$, $r_1\neq r_2$. Alla hommogena lösningar ges av*$$C_1e^{r_1x}+C_2e^{r_2x}$$*Fall 2: Karakteristiska polynomet har reel dubbelrot $r_0$. Alla homogena lösningar ges av*$$\left(C_1x+C_2\right)e^{r_0x}$$*Fall 3: Karakteristiska polynomet har komplexa rötter $k+i\omega$.*$$\left(A\sin\omega{x}+B\cos\omega{x}\right)e^{kx}$$
+	- **Ex Homohena**$$\begin{align}\text{Fall 1: }y''-3y'+2y=0\\\text{Karakteristiska polynomet}\\P(n)=n^2-3n+2\\P(n)=0\Leftrightarrow\left(n-2\right)\left(n-1\right)\\\Leftrightarrow{n}=1\text{ eller }2\\y_h=C_1e^xĆ_2e^{2x}\\\\\text{Fall 2: }y''-4y'+4y=0\\P(n)=n^2-4n-4=\left(b-2\right)^2\\P(n)=0\Leftrightarrow\left(n-2\right)^2=0\Leftrightarrow n=2\\y_h=\left(C_1c+C_0\right)e^{2x}\\\\\text{Fall 3: }y''-4y+5y=0\\P(n)=n^2+4n+5=\left(n-2\right)^2+1\\P(n)=0\Leftrightarrow\left(n-2\right)^2+1=0\\\Leftrightarrow n=2\pm i\end{align}$$
+	- **Ansatser**
+		- $h(x)=P(x)\Rightarrow y_p(x)=x^mA(x),\;grad(A)=grad(1).$
+		  **Ex**: $h(x)=x^2\Rightarrow y_p(x)=x^m\left(a_2x^2+a_1x+a_0\right)$ $$\begin{align}y''-3y'+2y=x^2+1\\y_p=ax²+bx+c\\\Rightarrow y_p'=2ax+b\\\Rightarrow y_p''=2a\\\text{Sätt in i ODE}\\3a-3\left(2ax+b\right)+2\left(ax^2+bx+c\right)=x^2+1\\\Leftrightarrow 2ax^2+\left(2b-6a\right)x+2a-3b+2c=x^2+1\\\text{Jämför koeffieinten:}\\x^2:\;\;2a=1\Leftrightarrow a=\frac12\\x^2:\;\;2b-6a=0\Leftrightarrow b=3a=\frac32\\x^0:\;\;2a-3b+2x=1\Leftrightarrow2x=1-2a+3b=1-1+\frac92\\\Leftrightarrow c=\frac94\\\underline{\text{sum}}:\;y_p==\frac12x^2+\frac32x+\frac94\\\text{Almän lösning till ODE:}\\y=t_h+y_p=C_1e^x+C_2e^{2x}+\frac12x^2+\frac32x+\frac94\end{align}$$$$\begin{align}y''=x+1\\y_h=C_x+C_0\\y_p=x^2\left(ax+b\right)=ax^3+bx^2\\\Rightarrow y'_p=3ax^2+2bx\\\Rightarrow y''_p=6ax+2b\\\text{Sätt in }y_p\text{ i ODE: }y''_p=x+1\\\Leftrightarrow6ax+2b=x+1\\\Leftrightarrow6a=1,2b=1\Leftrightarrow a=\frac16,b=\frac12\\\underline{\text{Svar}}:\;y=\frac16x^3\frac12x^2+C_1x+C_0\end{align}$$
+		- 
 - **Examples**
 	- $$\begin{align}y^2y'=2xy^{1/2}\\\text{Lösn: För }y(x)\neq0,\\y^2y'=2xy^{1/2}\Leftrightarrow y^{3/2}y'=2x\\\text{Integrera m.a.p. }x,\\\frac25y^{2/5}=x^2+C\Leftrightarrow C=\frac25-1=-\frac35\\\text{Lösning är}\\y\begin{aligned}=\left(\frac52\left(x^2-\frac35\right)\right)^{2/5}\\=\left(\frac52x^2-\frac32\right)^{2/5}\end{aligned}, x^2\geq\frac35\\x\leq\sqrt{-\frac35}\text{ eller }x\geq\sqrt\frac35\end{align}$$
 	- $$\begin{align}e^{x^2}+y'e^{x^2}\times2xy=\left(e^{x^2}y\right)'\end{align}$$