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5 Cards in this Set
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Given a general differential equation in standard form, dy/dx = ay  b, and given the initial condition y(0) = y_0, find the general solution and solve the initial value problem. Also find the parameter value(s) corresponding to the equilibrium solution.

general solution: y = b/a + ce^(at), a != 0, c = +or e^C
particular solution: y = b/a + [y_0  b/a]e^(at) equilibrium solution: y = b/a, c= 0 

The geometric representation of the general solution is an infinite family of curves called ________. Each one is associated with a particular ________.
Satisfying an initial condition amounts to ________. 
integral curves
cvalue finding the integral curve that corresponds to the given initial point 

Solve the general predatorprey (owlmouse) problem given p(0) = p_0.

p = k/r + [p_0  k/r]e^(rt)


Explain the falling object and owlmouse models in terms of the general equation.

owlmouse:
a = gamma/m b= g v = gm/gamma + [v_0  gm/gamma]e^(tgamma/m) As t increases, the second term dies off and we converge to equilibrium. The higher the drag coefficient relative to the mass of the object, the faster this occurs. owlmouse: a = k b = r p = k/r + [p_0  k/r]e^(rt) If the population begins at the equilibrium value, it stays. Otherwise, the second term grows exponentially and the mouse population either dies off or grows infinitely. 

Consider an electric circuit containing a capacitor, resistor, and battery. The Charge Q(t) on the capacitor satisfies the equation
RdQ/dt + Q/C = V where R is the resistance, C is the capacitance, and V is the constant voltage supplied by the battery. a)If Q(0) = 0, find Q(t) at any time t. b)Find the limiting value Q_L that Q(t) approaches after a long time. c) Suppose that Q(t_1) = Q_L and that the battery is removed from the circuit at t = t_1. Find Q(t) for t > t_1. 
a) Q(t) = CV(1  e^(t/CR))
b)Q_L = lim t>inf Q(t) = CV c)Q(t) = CVe^(tt_1)/CR 