different terms revolving around differential equations
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
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| ordinary differential equations | equations involving a function and its derivatives
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| order | the order of the highest derivative
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| general solution | y=mt+b since we don't know the values for m and b
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| particular solution | a solution with a specific value for c, from an initial value problem
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| interval of existence | a set of t-values for which the solution x(t) is defined
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| 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
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| autonomous | if the RHSde has no explicit t's
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| homogeneous | if q(t)=0 in an ode
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| linear ode formula | dx/dt=p(t)*x+ q(t)
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| integrating factor | used to write the LHS as the derivative of the product of itself and the dependent variable x
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| law of cooling | dx/dt= a(x(t)-b) and soln: x(t)=b+(x0-b)*e^(-at)
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| exponential decay | dx/dt=-r*x and soln: x(t)=x0*e^(-rt)
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| exponential growth | dx/dt=-r*x and soln: x(t)=x0*e^(rt)
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| general form for nonhomogeneous ode | x=xh+xp assuming p is constant
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| critical point/equilibrium | in an autonomous ode set dx/dt=0 to find this
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| 3 types of equilibriaugh | sink, source, shunt
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| sink | if f is decreasing through xe, also if f'(xe)<0
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| source | if f is increasing through xe, also if f'(xe)>0
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| shunt | if f has a local min or max at xe
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| nonhyperbolic equilibrium | if f'(xe)=0
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| hyperbolic equilibrium | if f'(xe)does not =0, this is easier to determine behavior from
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| linearization | way of studying solutions of nonlinear x'=f(x) near a hyperbolic equilibrium xe
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| 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)
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