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Unformatted text preview: (2 cos 2π t )P + 2000 cos 2π t dt describes a population that undergoes periodic ﬂuctuations as well as periodic migration. Continue to assume that P (0) = 1000 and ﬁnd P (t ) in this case. Use a graphing utility to draw the graph of P and estimate the maximum value of P . c 46. The Gompertz equation dP = P (a − b ln P ), dt where a and b are positive constants, is another model of population growth. (a) Find the solution of this differential equation that satisﬁes the initial condition P (0) = P0 , where P0 > 0. HINT: Deﬁne a new dependent variable Q by Q = ln P . (b) Find lim P (t ).
t →∞ c 45. (a) The differential equation
dP = (2 cos 2π t )P dt (c) Determine the concavity of the graph of P . (d) Use a graphing utility to draw the graph of P in the case where a = 4, b = 2, and P0 = 1 e2 . Does the graph 2 conﬁrm your result in part (c)? ∗ 8.9 INTEGRAL CURVES; SEPARABLE EQUATIONS Integral Curves
If a function y = y(x) satisﬁes a ﬁrst-order differential equation, then along the graph of the function, the numbers x, y, y are related as prescribed by the equation. In the case of a nonlinear differential equation, it is usually difﬁcult (and sometimes impossible) to ﬁnd functions which are explicitly deﬁned and satisfy the differential equation. More often, we can ﬁnd plane curves which, though not the graphs of functions, do have the property that along the curve the numbers x, y, y are related as prescribed by the differential equation. Such curves are called integral curves (solution curves) of the differential equation. To solve a nonlinear differential equation is to ﬁnd the integral curves of the equation. Separable Equations
A ﬁrst-order differential equation is said to be separable if it can be written in the form
(8.9.1) p(x) + q(y)y = 0. The functions p and q are assumed to be continuous where deﬁned....
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.
- Spring '10