2 obtain linearly independent solutions u1 u2 ur

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Unformatted text preview: = 1 2 : : : r: (7.1.8) Again, we'll assume that a means of nding u( ) , i = 1 2 : : : r, to satisfy the initial condition (7.1.1b) can be found. With this, we observe that the work of solving this system is about half that of the previous procedure (7.1.2c) when the number of terminal conditions r is approximately m=2. i 3. Write the general solution of the linear BVP (7.1.1) as y(x) = c1u(1) (x) + c2u(2) (x) + : : : + cr u(r) (x) + v(x) or y(x) = U(x)c + v(x) (7.1.9a) where U(x) = u(1) (x) u(2) (x) : : : u(r) (x)] 6 c = c1 c2 : : : cr ]T : (7.1.9b) 4. By construction, this representation satis es the ODE (7.1.1a) and the initial conditions (7.1.1b). It remains to satisfy the terminal conditions (7.1.1c) and this can be done by determining c so that Ry(b) = R U(b)c + v(b)] = r or RU(b)c = r ; Rv(b): (7.1.10) As noted, the main savings of this procedure relative to (7.1.2-7.1.3) is the reduced number of ODEs that have to be solved. This is o set by (possible) di culties associated with select...
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