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136 lines
2.4 KiB
136 lines
2.4 KiB
\documentclass{article}
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\usepackage[margin=2cm]{geometry}
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\usepackage{titlesec}
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\usepackage{titling}
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\usepackage[hidelinks]{hyperref}
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\usepackage{multicol}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\renewcommand{\maketitle}{
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\begin{center}
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{\huge\bfseries\thetitle}\\
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{\Large\theauthor} \\
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\vspace{1em}
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elalmqvist@gmail.com --- \url{https://wych.dev}
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\end{center}
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}
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\begin{document}
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\title{Anteckningar 2022-04-28}
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\author{Elias Almqvist}
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\maketitle
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\newpage
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Lös ekvationen:
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$$
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y'' = -2y' - 8y + \sin x
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$$
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Lösning:
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$$
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y'' = -2y' - 8y + \sin x, \quad + 2y' + 8y
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$$
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$$
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y'' + 2y' + 8y = \sin x
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$$
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Vi får därmed att $y = y_p + y_h$, \\
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Börjar med den homogena:
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$$
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y'' + 2y' + 8y = 0
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$$
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$$
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\implies r^2 + 2r + 8 = 0 \quad \text{antag att y är } e^{rx}
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$$
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$$
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\implies r = -1 \pm \sqrt{-7} = -1 \pm 7i
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$$
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$$
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\therefore y_h = e^{-x} \left( A\cos(\sqrt{7}x) + B\sin{\sqrt{7}x} \right) \quad | \quad A,B \in \mathbb{R}
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$$
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Part lösn: antar att $y_p$ är av formen: $y_p = D\sin x + E\cos x + F \quad | \quad D,E,F \in \mathbb{R}$ \\
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Stoppar in den i ekvationen:
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$$
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y_p' = D\cos x - E\sin x
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$$
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$$
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y_p'' = -\left( D\sin x + E\cos x \right)
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$$
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Vi får därmed:
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$$
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-\left( D\sin x + E\cos x \right) + 2\left(D\cos x - E\sin x\right) + 8\left(D\sin x + E\cos x + F\right) = \sin x
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$$
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$$
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-D\sin x - E\cos x + 2D\cos x - 2E\sin x + 8D\sin x + 8E\cos x + 8F = \sin x
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$$
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$$
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2D\cos x - 2E\sin x + 7D\sin x + 7E\cos x + 8F = \sin x
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$$
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$$
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\implies
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\begin{cases}
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8F = 0 \implies F = 0 \\
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7E + 2D = 0 \\
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7D - 2E = 1
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\end{cases}
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$$
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Stoppar in i en matris och får:
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$$
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\begin{bmatrix}
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7 & 2 & 0 \\
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-2 & 7 & 1
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\end{bmatrix}
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\rightarrow
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\begin{bmatrix}
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1 & 0 & -\frac{2}{53} \\
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0 & 1 & \frac{7}{53}
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\end{bmatrix}
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$$
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$$
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\therefore y_p = \frac{7}{53} \sin x - \frac{2}{53} \cos x
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\quad \because E = -\frac{2}{53}, \quad D = \frac{7}{53}
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$$
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Slutligen får vi därmed hela allmäna lösningen:
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$$
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\therefore y = y_p + y_h
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$$
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$$
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= \frac{7}{53} \sin x - \frac{2}{53} \cos x + e^{-x} \left( A\cos(\sqrt{7}x) + B\sin{\sqrt{7}x} \right)
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$$
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Givet att $A,B \in \mathbb{R}$
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\end{document}
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