MEC702_Wk_7_Lect_1

# 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 Coecients or Extended Method of Variation of Parameters. 1 Extended Method of Undetermined Coecients 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 coecients by substituting yp and its derivatives into the ODE. 2. Modication 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 Modication 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 coecients, 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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• Winter '16
• Alveen Chand
• Trigraph, yp, 6y 00 + 12y 0 + 8y = 8x2

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