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Calculus Advanced /
Ordinary Differential Equations.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} {\large Ordinary Differential Equations (ODE)} \begin{align*} \text{\bf General }&\text{\bf Form:}\quad\sum_{k=0}^nf_k(x,y)\frac{d^ky}{dx^k}=G(x,y).\quad\left(\frac{d^0y}{dx^0}\equiv y\right).\\ &\text{For convenience, it is usually reduced to $f_n(x,y)=1$. This is implied in this text.}\\ \\ \text{\bf First-}&\text{\bf Order ODEs:\rm\quad$\frac{dy}{dx}=F(x,y),\quad\big($i.e. $F(x,y)=G(x,y)-f_0(x,y)y\big).$}\\ &\text{Separable:\quad$\frac{dy}{dx}=f(x)g(y)$.\qquad Solution: $\int\frac{dy}{g(y)}=\int f(x)dx+c$.}\\ &\text{Linear:\quad$\frac{dy}{dx}+P(x)y=Q(x).\quad\big($i.e. $f_0(x,y)=P(x),~~G(x,y)=Q(x),~~$both independent of $y.\big)$}\\ &\qquad\text{Integrating factor $I=e^{\int P(x)dx}$.\qquad Solution: $\frac{d}{dx}\left(Iy\right)=I\frac{dy}{dx}+IP(x)y=IQ(x),~~y=\frac{1}{I}\left(\int IQ(x)dx+c\right)$.}\\ \\ \text{\bf Second-}&\text{\bf Order ODEs:\quad$\frac{d^2y}{dx^2}+f_1(x,y)\frac{dy}{dx}+f_0(x,y)y=G(x,y).$}\\ &\text{Homogeneous \big($G(x,y)=0$\big) with Constant Coefficients:\quad$\frac{d^2y}{dx^2}+a\frac{dy}{dx}+by=0.$}\\ &\quad\text{Solutions form a 2-dimensional vector space:\quad$y=\alpha_1y_1(x)+\alpha_2y_2(x)$,}\\ &\qquad\text{where $y_1$ and $y2$ are linearly independent. If they are linearly dependent, $y=(\alpha_1+\alpha_2x)y_1(x)$.}\\ &\quad\text{Function $y=e^{\lambda x}$ will make $\lambda^2e^{\lambda x}+a\lambda e^{\lambda x}+be^{\lambda x}=0.\quad\lambda^2+a\lambda+b=0$.}\\ &\quad\text{For two distinct real roots: $y=\alpha_1e^{\lambda_1x}+\alpha_2e^{\lambda_2x}$.\quad For two equal real roots $\left(\lambda=\frac{-a}{2}\right)$: $y=(\alpha_1+\alpha_2x)e^{\frac{-ax}{2}}$.}\\ &\quad\text{For two complex roots }\left(\frac{-a}{2}\pm iw\text{, where }w=\sqrt{b-\frac{a^2}{4}}\right)\text{: }y=\alpha_1e^{(\frac{-a}{2}+iw)x}+\alpha_2e^{(\frac{-a^2}{2}-iw)x}.\\ &\quad y=e^{\frac{-ax}{2}}\left(\alpha_1e^{iwx}+\alpha_2e^{-iwx}\right) =e^{\frac{-ax}{2}}\left(\alpha_1(\cos(wx)+i\sin(wx))+\alpha_2(\cos(wx)-i\sin(wx))\right).\\ &\quad y=e^{\frac{-ax}{2}}\left(\beta_1\cos(wx)+\beta_2\sin(wx)\right),\quad\text{where }\beta_1\text{ and }\beta_2\text{ are two arbitrary complex numbers}.\\ \\ &\text{Free Oscillations:\quad Force~=~-~friction~-~elasticity,~~$ma=-cv-ky$,\quad or\quad$m\frac{d^2y}{dt^2}+c\frac{dy}{dt}+ky=0$,\quad$(m,c,k>0)$.}\\ &\quad\text{Overdamping (large friction $c^2>4mk$): $y=Ae^{\lambda_1t}+Be^{\lambda_2t}.\quad\lim_{t\to\infty}y=0.$}\\ &\quad\text{Critical damping ($c^2=4mk$): $y=(A+Bt)e^{-\frac{ct}{2m}}.\quad\lim_{t\to\infty}y=0.$}\\ &\quad\text{Underdamping ($c^2<4mk$): $y=\left(A\cos(\Omega t)+B\sin(\Omega t)\right)e^{-\alpha t}$,\quad where $\alpha=\frac{c}{2m}$ and $\Omega=\sqrt{\frac{k}{m}-\frac{c^2}{4m^2}}$.}\\ &\qquad\text{For a right angle triangle with sides $A$, $B$, and angle $\delta$ between $R$ and $A$, $R=\sqrt{A^2+B^2}$ and $\tan(\delta)=\frac{B}{A}$.}\\ &\qquad y =R\left(\frac{A}{R}\cos(\Omega t)+\frac{B}{R}\sin(\Omega t)\right)e^{-\alpha t} =R\left(\cos(\delta)\cos(\Omega t)+\sin(\delta)\sin(\Omega t)\right)e^{-\alpha t}.\\ &\qquad\text{$y=Re^{-\alpha t}\cos(\Omega t-\delta),$ a decaying oscillation if $c>0$.}\\ &\qquad\quad\text{If $c=0$ (no friction), $y=R\cos(\Omega_0t-\delta)$, where $\Omega_0=\sqrt{\frac{k}{m}}$. The period is $\frac{2\pi}{\Omega_0}$.}\\ \end{align*} \end{document}