-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^(-
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
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