Unformatted text preview: Consider the ﬁrst-order equation dx dt = x 2-1 . (1) (a) Show that Z x 1 x dx x 2-1 = Z t 1 t dt (2) holds for all trajectories x ( t ) with x ( t ) = x and x ( t 1 ) = x 1 . (b) Suppose a trajectory starts at x = 2 at t = 0 . At what time does it reach x = ∞ ? (c) For a general initial value problem ˙ x = f ( x ) , x (0) = x , suppose f ( x ) > for all x ≥ x . Can you formulate a condition for x ( t ) → ∞ in ﬁnite time? Hint: You can state your answer in terms of an integral similar to that in Eq. (2)....
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- Spring '10
- Math, Trigraph, Boundary value problem, Sect., ﬁrst-order equation dx