MEC702_Wk_7_Lect_1 - MEC702 Wk 7 Lect 1 Non-homogeneous...

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Unformatted text preview: MEC702 Wk 7 Lect 1 Non-homogeneous Linear ODEs We now turn from homogeneous to non-homogeneous linear ODEs of nth order. We write them in standard form as y (n) + pn−1 (x)y (n−1) + · · · + p1 (x)y 0 + p0 (x)y = r(x) . Just like in Chapter 2, the solution of non-homogeneous ODE is of the form y = yh + yp where yh is the general solution for the homogeneous case and yp is the particular solution found for the non-homogeneous case. • yh is found using λ-method. • yp is found using Extended Method of Undetermined Coecients or Extended Method of Variation of Parameters. 1 Extended Method of Undetermined Coecients This method still includes the same table used in Chapter 2: Term in r(x) γx ke kx , (n = 0, 1, . . .) k cos ωx k sin ωx keαx cos ωx n Choice for yp (x) Ceγx Kn x + Kn−1 x + · · · + K1 x + K0 keαx sin ωx n n−1 K cos ωx + M sin ωx eαx (K cos ωx + M sin ωx) The rules given in Chapter 2: 1. Basic Rule. If r(x) in ODE is one of the functions in the rst column in the Table below, choose yp in the same line and determine its undetermined coecients by substituting yp and its derivatives into the ODE. 2. Modication Rule. If a term in your choice for yp happend to be a solution of the homogeneous ODE corresponding to the nonhomogeneous ODE, multiply this term by x (or x2 if this solution corresponds to a double root of the characteristic equation of the homogeneous ODE.) 3. Sum Rule. If r(x) is a sum of functions in the rst column of the Table below, choose for yp the sum of the functions in the corresponding lines of the second column. The rules for the extended method: 1. Extended Basic Rule: same as the one given in Chapter 2. 2. Extended Sum Rule: same as the one given in Chapter 2. 3. Extended Modication Rule: If the term y∗ in your choice of yp (x) is a solution of the homogeneous case, then multiply this term by xk to obtain xk y∗, where k is smallest positive integer such that xk y∗ is not a solution of the homogeneous case. Example. Solve the IVP y 000 + 3y 00 + 3y 0 + y = 30e−x , y(0) = 3, y 0 (0) = −3, Solution. The particular solution of the IVP is y = (3 − 25x2 )e−x + 5x3 e−x . Exercise. Solve the ODE y 000 + 6y 00 + 12y 0 + 8y = 8x2 . 2 y 00 (0) = −47 . Extended Method of Variation of Parameters The method of variation of Parameters given in Chapter 2 is given by ˆ yp (x) = −y1 y2 r dx + y2 W ˆ y1 r dx W Now, extending this method to nth order non-homogeneous ODEs with constant coecients, we have  ˆ n  X Wk (x) yp (x) = yk (x) r(x)dx W (k) k=1 or ˆ  yp (x) = y1 (x)      ˆ ˆ W1 (x) W2 (x) Wn (x) r(x)dx + y2 (x) r(x)dx + · · · + yn (x) r(x)dx W (x) W (x) W (x) Example. Solve the Euler-Cauchy equation x3 y 000 − 3x2 y 00 + 6xy 0 − 6y = x4 ln x, Solution. We have (x > 0) . yh = c1 x + c2 x2 + c3 x3 where the basis of solutions are y1 = x, y2 = x2 , y3 = x3 . Then W Then = 2x3 W1 = x4 W2 = −2x3 W3 = x2 .   11 1 4 . yp = x ln x − 6 6 Exercise. Solve the ODE y 000 + 6y 00 + 12y 0 + 8y = 8x2 . 3 ...
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