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# Math 312 test 1

### different terms revolving around differential equations

ordinary differential equations equations involving a function and its derivatives
order the order of the highest derivative
general solution y=mt+b since we don't know the values for m and b
particular solution a solution with a specific value for c, from an initial value problem
interval of existence a set of t-values for which the solution x(t) is defined
existence and uniqueness theorem if f(t,x) and its derivative are continuous near x0 and t0 then the IVP has one and only one solution
autonomous if the RHSde has no explicit t's
homogeneous if q(t)=0 in an ode
linear ode formula dx/dt=p(t)*x+ q(t)
integrating factor used to write the LHS as the derivative of the product of itself and the dependent variable x
law of cooling dx/dt= a(x(t)-b) and soln: x(t)=b+(x0-b)*e^(-at)
exponential decay dx/dt=-r*x and soln: x(t)=x0*e^(-rt)
exponential growth dx/dt=-r*x and soln: x(t)=x0*e^(rt)
general form for nonhomogeneous ode x=xh+xp assuming p is constant
critical point/equilibrium in an autonomous ode set dx/dt=0 to find this
3 types of equilibriaugh sink, source, shunt
sink if f is decreasing through xe, also if f'(xe)<0
source if f is increasing through xe, also if f'(xe)>0
shunt if f has a local min or max at xe
nonhyperbolic equilibrium if f'(xe)=0
hyperbolic equilibrium if f'(xe)does not =0, this is easier to determine behavior from
linearization way of studying solutions of nonlinear x'=f(x) near a hyperbolic equilibrium xe
separation of variables can be used for all autonomous and some nonautonomous odes if x'=f(t,x) can be separated to f(t,x)=g(t)*h(x)
Created by: lfalkens