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发表于 22-7-2013 09:03 PM
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A differential equation has a family of solutions with different shapes of solution curves. It is interesting to investigate the shape of the solution curves.
We shall investigate the shapes of the solution curves for the first order differential equation
dy/dx = ax^(2) + bx + c, where a, b and c are constants and a ≠ 0
1. The direction field of a differential equation dy/dx = f(x,y) is a collection of short line segments through each of a sequence of point (x,y) with slope f(x,y) respectively. Thus, the direction field allows us to visualise the general shapes of the solution curves.
=====By sketching the direction field of the differential equation dy/dx = ax^(2) +bx +c, investigate the shapes of the solution curves for various values of a, b and c.
2.(a)find the general solution of the differential equation dy/dx = ax^(2) +bx +c
(b)sketch the solution curves passing through the origin for different values of a, b and c.
3.(a)determine the conditions for a solution curve
(i) to be increasing and have no turning point
(ii) to be decreasing and have no turning point
(iii) to have turning point
(b) find the range of values of x, in terms of a, b and c, for a solution curve to be increasing and decreasing. |
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