chapter2.page16

chapter2.page16 - If f x> 0 the graph is increasing If f...

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16 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS is x ( t ) = t 1 3 . The reason is that for f ( t,x ) = x 2 3 , ∂f ( t,x ) ∂x = 1 3 x which is undeﬁned at (0 , 0) . So the conditions of the theorem are violated. 5. Phase and vector ﬁeld diagrams Sometimes it is more important to know the behavior of solutions than to know precisely detail of point-by-point of some particular so- lutions. Here we will discuss two diagrams to get an ”feel” of behavior of solutions for a ODE without actually ﬁnd any solution. The ﬁrst, phase diagram , is used for the autonomous equations and is easy to draw. The second, vector ﬁeld diagram , can be used for any equa- tion but harder to draw. These two diagrams make use of the facts: (1) The derivative f 0 ( x ) represent the slope of tangent line to the graph y = f ( x ) at the point ( x,f ( x )) . (2) The sign of f 0 ( x ) tell whether the graph is going up or down.
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Unformatted text preview: If f ( x ) > 0 the graph is increasing. If f ( x ) < 0 the graph is going down. (3) The second derivative f 00 ( x ) gives information about concav-ity: If f 00 ( x ) > 0 the graph is concave upward. If f 00 ( x ) < 0 the graph is concave downward. The concepts of concavity (up or down) and monotonicity (going up or down) is illustrated in the following diagram. Figure 8. Concavity and Monotonicity 5.1. The phase diagram. First-order equations whose right-hand side does not depend on the independent variable are called autonomous. That is, autonomous equations are those of the form x = dx dt = f ( x ) (The independent variable is t )or y = dy dx = f ( y ) (The independent variable is x ). To draw a phase diagram of x = f ( x ), you would...
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