Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The substitution y eAx yields A2 - 2A 4 A - 2 2 0. Because the polynomial has a double root the solutions are e2x and x e2x. Result Consider the second order constant coefficient differential equation y 2ay by 0. We can factor the differential equation into the form Ặ - a - fi y 0 dx dx which has the solution f C1 eax c2 e x a fi y I Ci eax c2x eax a fi. We can also determine a and fi by substituting y eXx into the differential equation and factoring the polynomial in A. Shift Invariance. Note that if u x is a solution of a constant coefficient equation then u x c is also a solution. This is useful in applying initial or boundary conditions. Example Consider the problem y - 3y 2y 0 y 0 a y 0 b. We know that the general solution is y C1 C2 e2x . 934 Applying the initial conditions we obtain the equations Ci C2 a Ci 2C2 b. The solution is y 2a b ex b a e2x . Now suppose we wish to solve the same differential equation with the boundary conditions y T a and y T b. All we have to do is shift the solution to the right. y 2a b ex-1 b a e2 x-1 . Real-Valued Solutions If the coefficients of the differential equation are real then the solution can be written in terms of real-valued functions Result . For a real root A a of the polynomial in A the corresponding solution y eax is real-valued. Now recall that the complex roots of a polynomial with real coefficients occur in complex conjugate pairs. Assume that a ift are roots of An an-iAn x aiA a0 0. The corresponding solutions of the differential equation are e a 1 x and e a-1 x. Note that the linear combinations e a i x I e a-I x e a i x e a-I0 x are real-valued solutions of the differential equation. We could also obtain real-valued solution by taking the real and imaginary parts of either e a 1T x or e a-1T x. e x eax cos Ar s e a 1 x eax sin x Example Consider the equation y 2y 2y 0. .
