Mathematics MATH 511 Chapter Notes -Integrating Factor, Terrestrial Trunked Radio
7 Dec 2013
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A differential equation is any equation which involves a function y(x) and any of its derivatives y", y"", y""", . It can be as easy as y"=2 to y"=xy to y""-xy"=ln(y) to awful things like cos(xyy")=(e^(- x^2y"))/(y^3arctan(lnlyl)). A population of 2000 bacteria grows at a rate of 3% per hour for the next 24 hours. Find the population size after 12 hours. y"=ky y=e^(kx) A linear equation means the exponent only is 1. Growth rate = 3% per hour p(t) = population after t hours. P"(t) = . 03 p(t) p(t) = ce^(. 03t) The first class of differential equations are first order linear equations. 1st order linear differential equation has the form y"(x)+p(x)y=q(x) If q(x)=0, the equation is called homogenous. y"=xy guess y=ce^(x*x)=ce^(x^2) y=ce^(x^2) Once you calculate the integrating factor, multiply through the entire equation. Once you calculate the integrating factor, multiply through the entire equation. y"+p(x)y=q(x) e^(int. p(x)dx)y"+e^(int. p(x)dx)p(x)y=e^(int. p(x)dx)q(x) p(x)=x, q(x)=0 y"=(ce^(x^2))(2x)=2xce^(x^2)-2xy y"=(ce^(x^2/2))(x)=x(ce^(x^2/2)) (e^(int. p(x)dx)y)"=e^(int. p(x)dx)q(x)
