62 lines
1.6 KiB
TeX
62 lines
1.6 KiB
TeX
% File: diffEqReview.tex
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% Created: 12:36:05 Fri, 27 Aug 2021 EDT
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% Last Change: 12:36:05 Fri, 27 Aug 2021 EDT
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%
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\documentclass[letterpaper]{article}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{cancel}
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\usepackage{amssymb}
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\usepackage{listings}
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\usepackage{enumitem}
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\date{08/27/2021}
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\title{%
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Differential Equations Overview\\
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\large EEET-427-01: Control Systems}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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\section{General Steps}
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Given $F(x)=G(x)$
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\begin{enumerate}
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\item Solve for roots of function
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\item Using roots, fill in general solution of $F(x)$ (assuming $G(x)=0$). General solution options:
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\begin{itemize}
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\item Real roots ($c_1e^{r_1x}+c_2e^{r_2x}\dots$)
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\item repeated roots ($c_1e^{r_1x}+c_2xe^{r_1x}\dots$)
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\item imaginary roots ($e^{\alpha x}(c_1\cos(\beta x)+c_2\sin(\beta x))$)
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\end{itemize}
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\item Use initial conditions (if given) to find $c$ values.
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\item Generalize $G(x)$, take necessary derivatives, and plug in to $F(x)$, solve for coefficients
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\end{enumerate}
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\section{Example}
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\begin{equation}
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\begin{split}
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y''+5y'+6y=x^2\\
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y''+5y'+6y=0\\
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r^2+5r+6=0\\
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r=-2,\ -3\\
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y_g(x)=c_1e^{-2x}+c_2e^{-3x}\\
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G(x)=x^2\\
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y_p(x)=Ax^2+Bx+C\\
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y'(x)=2Ax+B\\
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y''(x)=2A\\
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y''+5y'+6y=x^2\\
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2A+5(2Ax+B)+6(Ax^2+Bx+C)=x^2\\
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2A+10Ax+5B+6Ax^2+6Bx+6C=x^2\\
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6Ax^2+(10A+6B)x+(2A+5B+6C)=x^2+0x+0\\
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6A=1\\
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10A+6B=0\\
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2A+5B+6C=0\\
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A=\frac16\\
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B=-\frac{10}{36}\\
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C=\frac{38}{216}\\
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y_p(x)=\frac16x^2-\frac{10}{36}x+\frac{38}{216}\\
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y(x)=\frac16x^2-\frac{10}{36}x+\frac{38}{216}+c_1e^{-2x}+c_2e^{-3x}
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\end{split}
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\end{equation}
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\end{document}
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