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Math 312 test 1
different terms revolving around differential equations
| Question | Answer |
|---|---|
| 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
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