master
E. Almqvist 3 years ago
parent 71be35d0b0
commit e5b047071e
  1. 19
      ma5/notes/test/test.aux
  2. 85
      ma5/notes/test/test.fdb_latexmk
  3. 553
      ma5/notes/test/test.fls
  4. 400
      ma5/notes/test/test.log
  5. 0
      ma5/notes/test/test.out
  6. BIN
      ma5/notes/test/test.pdf
  7. BIN
      ma5/notes/test/test.synctex.gz
  8. 136
      ma5/notes/test/test.tex

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