[ykang]Differential Equations note.docx

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Mathematics MATH 511
van Groningen

Differential Equations -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 common example is population growth. 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. Growth rate = 3% per hour P(t) = population after t hours P’(t) = .03 P(t)  P(t) = ce^(.03t) P(0)=2000 c=2000 Given one point, called an initial condition P(t) 2000 e^(.03t) P(12) = 2000e^(.03*12) = 2867 y’=ky y=e^(kx) The first class of differential equations are first order linear equations. First order means you only have y’ A linear equation means the exponent only is 1 st 1 order linear differential equation has the form y’(x)+p(x)y=q(x) One example is y’=xy  y’-xy=0 p(x)=x, q(x)=0 If q(x)=0, the equation is called homogenous. y’=xy Guess y=ce^(x*x)=ce^(x^2) y=ce^(x^2) y’=(ce^(x^2))(2x)=2xce^(x^2)-2xy
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