14 A Differential Equation Has Integral Curves X 21 C A Sketch Several Of These Curves In The Ta Plane B Fin 1 (53.77 KiB) Viewed 14 times
14 A Differential Equation Has Integral Curves X 21 C A Sketch Several Of These Curves In The Ta Plane B Fin 2 (53.77 KiB) Viewed 14 times
14. A differential equation has integral curves x² +21² = C. (a) Sketch several of these curves in the ta plane. (b) Find the differential equation. (c) Find the explicit solution satisfying the condition t = 1, x = 4 and determine its interval of existence. 15. Find the general solution of the DE x' = 6t(x - 1)2/3. Clearly, r(t) = 1 is a constant solution. Solve the DE and show that there is no value of the arbitrary constant that gives the solution z(t) = 1. (A solution to a DE that cannot be obtained from the general solution by fixing a value of the arbitrary constant is called a singular solution.) 16. (Mechanics) We modeled the velocity of an object falling in a fluid by the equation mv' = mg-av². Use separation of variables and partial fractions to show that the general solution is v(t) If v(0) V, determine the constant C and then find the limiting, or ter- minal, velocity. rm (et-Ce-t- a ert + Ce-rt x(K 6-7). 17. Find the general solution to the logistic equation z' = ræ(1 a/K) using separation of variables. Hint: use the partial fractions decomposition 1 x) 1/K 1/K ag m + I K-x'
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