mirror of https://github.com/E-Almqvist/hsf
parent
71be35d0b0
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\relax |
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\providecommand\hyper@newdestlabel[2]{} |
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\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} |
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\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined |
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\global\let\oldcontentsline\contentsline |
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\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} |
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\global\let\oldnewlabel\newlabel |
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\gdef\newlabel#1#2{\newlabelxx{#1}#2} |
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\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} |
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\AtEndDocument{\ifx\hyper@anchor\@undefined |
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\let\contentsline\oldcontentsline |
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\let\newlabel\oldnewlabel |
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\fi} |
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\fi} |
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\global\let\hyper@last\relax |
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\gdef\HyperFirstAtBeginDocument#1{#1} |
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\providecommand\HyField@AuxAddToFields[1]{} |
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\providecommand\HyField@AuxAddToCoFields[2]{} |
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\gdef \@abspage@last{2} |
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# Fdb version 3 |
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["pdflatex"] 1651137953 "test.tex" "test.pdf" "test" 1651137954 |
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"/usr/share/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1647844622 940 228d6584342e91276bf566bcf9716b83 "" |
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"/usr/share/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1647844622 1288 655e228510b4c2a1abe905c368440826 "" |
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"/usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1647844622 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" |
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"/usr/share/texmf-dist/tex/generic/pdfescape/pdfescape.sty" 1647844622 19007 15924f7228aca6c6d184b115f4baa231 "" |
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"/usr/share/texmf-dist/tex/latex/amsfonts/umsb.fd" 1647844622 961 d02606146ba5601b5645f987c92e6193 "" |
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"/usr/share/texmf-dist/tex/latex/amsmath/amsgen.sty" 1647844622 4173 bc0410bcccdff806d6132d3c1ef35481 "" |
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"/usr/share/texmf-dist/tex/latex/amsmath/amsmath.sty" 1647844622 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c "" |
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"/usr/share/texmf-dist/tex/latex/hyperref/puenc.def" 1647844622 117004 ed1c2cc82bb9836e9d59549dd8c33098 "" |
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"/usr/share/texmf-dist/web2c/texmf.cnf" 1647844622 39911 2da6c67557ec033436fe5418a70a8a61 "" |
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"test.aux" 1651137954 686 4ce905362b6645b598ea182482e9aeed "pdflatex" |
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"test.out" 0 -1 0 "pdflatex" |
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"test.tex" 1651137952 2451 4343002642170c6be9a13ae085881530 "" |
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(generated) |
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"test.aux" |
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"test.log" |
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"test.out" |
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"test.pdf" |
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PWD /home/elal/Projects/school/prog2/ma5/notes/test |
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INPUT /usr/share/texmf-dist/web2c/texmf.cnf |
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INPUT /var/lib/texmf/web2c/pdftex/pdflatex.fmt |
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INPUT test.tex |
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OUTPUT test.log |
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INPUT /usr/share/texmf-dist/tex/latex/base/article.cls |
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) (/usr/share/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty |
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Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO) |
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) |
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\c@section@level=\count295 |
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LaTeX Info: Redefining \ref on input line 46. |
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LaTeX Info: Redefining \pageref on input line 46. |
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(./test.out) (./test.out) |
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\@outlinefile=\write3 |
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\openout3 = `test.out'. |
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LaTeX Font Info: Trying to load font information for U+msa on input line 51. |
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(/usr/share/texmf-dist/tex/latex/amsfonts/umsa.fd |
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File: umsa.fd 2013/01/14 v3.01 AMS symbols A |
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) |
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LaTeX Font Info: Trying to load font information for U+msb on input line 51. |
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(/usr/share/texmf-dist/tex/latex/amsfonts/umsb.fd |
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File: umsb.fd 2013/01/14 v3.01 AMS symbols B |
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) [1 |
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{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] (./test.aux) |
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Package rerunfilecheck Info: File `test.out' has not changed. |
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(rerunfilecheck) Checksum: D41D8CD98F00B204E9800998ECF8427E;0. |
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) |
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Here is how much of TeX's memory you used: |
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9471 strings out of 478276 |
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144995 string characters out of 5853013 |
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451977 words of memory out of 5000000 |
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27584 multiletter control sequences out of 15000+600000 |
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472030 words of font info for 40 fonts, out of 8000000 for 9000 |
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1141 hyphenation exceptions out of 8191 |
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60i,7n,63p,221b,330s stack positions out of 5000i,500n,10000p,200000b,80000s |
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</usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb> |
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Output written on test.pdf (2 pages, 119551 bytes). |
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PDF statistics: |
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78 PDF objects out of 1000 (max. 8388607) |
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49 compressed objects within 1 object stream |
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\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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\usepackage{amssymb} |
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\usepackage{braket} |
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\titleformat{\section} |
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{\Large\bfseries} |
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{} |
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{0em} |
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{}[\titlerule] |
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\titleformat{\subsection} |
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{\large\bfseries} |
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{} |
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{0em} |
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{} |
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\titlespacing{\subsection} |
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{0em}{2em}{.4em} |
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\titleformat{\subsubsection}[runin] |
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{\bfseries} |
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{} |
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{0em} |
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{} |
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\titlespacing{\subsubsection} |
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{0em}{2em}{1em} |
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\renewcommand{\maketitle}{ |
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\begin{center} |
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{\huge\bfseries\thetitle}\\ |
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\vspace{1em} |
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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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|
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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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|
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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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|
|
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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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|
$$ |
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|
y_p' = D\cos x - E\sin x |
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|
$$ |
||||||
|
$$ |
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|
y_p'' = -\left( D\sin x + E\cos x \right) |
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|
$$ |
||||||
|
Vi får därmed: |
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|
$$ |
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|
-\left( D\sin x + E\cos x \right) + 2\left(D\cos x - E\sin x\right) + 8\left(D\sin x + E\cos x + F\right) = \sin x |
||||||
|
$$ |
||||||
|
$$ |
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|
-D\sin x - E\cos x + 2D\cos x - 2E\sin x + 8D\sin x + 8E\cos x + 8F = \sin x |
||||||
|
$$ |
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|
$$ |
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|
2D\cos x - 2E\sin x + 7D\sin x + 7E\cos x + 8F = \sin x |
||||||
|
$$ |
||||||
|
$$ |
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|
\implies |
||||||
|
\begin{cases} |
||||||
|
8F = 0 \implies F = 0 \\ |
||||||
|
7E + 2D = 0 \\ |
||||||
|
7D - 2E = 1 |
||||||
|
\end{cases} |
||||||
|
$$ |
||||||
|
|
||||||
|
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} |
||||||
Loading…
Reference in new issue