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Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 7

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | What would happen if we continued this method Since y x c1x is a solution of the Ricatti equation we can make the substitution 1.1 y x c x U X 1 which will lead to a solution for y which has two constants of integration. Then we could repeat the process substituting the sum of that solution and 1 u x into the Ricatti equation to find a solution with three constants of integration. We know that the general solution of a first order ordinary differential equation has only one constant of integration. Does this method for Ricatti equations violate this theorem There s only one way to find out. We substitute Equation 18.7 into the Ricatti equation. u 1 2x 2 I x ------ 1 u 1 2 1 u ---------u 1 2 u -------u 1 The integrating factor is I x e2 c-x e-2log c-x 1 2. Upon multiplying by the integrating factor the equation becomes exact. d 1 1 dx c x 2 c x 2 2 1 . 2 u c x 2 1 b c x 2 u x c b c x 2 Thus the Ricatti equation has the solution y x 1 1 c x x c b c x 2 1014 It appears that we we have found a solution that has two constants of integration but appearances can be deceptive. We do a little algebraic simplification of the solution. I 1 x c x b c x 1 c x b c x 1 1 y J b c x 1 c x b y x b c x 1 1 y x c 1 b x This is actually a solution namely the solution we had before with one constant of integration namely c 1 b . Thus we see that repeated applications of the procedure will not produce more general solutions. 3. The substitution u y T au gives us the second order linear homogeneous differential equation u j a b u acu 0. a The solution to this linear equation is a linear combination of two homogeneous solutions u1 and u2. u c1u1 x c2u2 x The solution of the Ricatti equation is then c1u 1 x c2u 2 x y a x ciui x c2u2 x . .