~ Y, f'li ~ &3 d4V\

The right-hand sides of Eqs. (8.117) and (8.121) are identical. Equating the left-hand sides of those equations gives

-"1 1 f~PI 2-"1 -"1) If:;. T - ) 0(&4)Y i+TIV' i+l - Y i+Y i-I = &2V'i+1 - Yi+Yi-1 + Define the second-order centered-differences J2Yi and J2y''I; as follows:

J2Yi = Yi+ I - 2Yi +Yi-l ~2-"1 _ -"1 _ 2-"1 + -"1U Y i - Y i+1 Y i Y i-I

Substituting Eqs. (8.123) and (8.124) into Eq. (8.122) gives J2-

-"I.+--1-J2-"1 =---.l!:+0(&4)Y, 12 Y, &2 Solving Eq. (8.125) for Y"li yields

J2--"1 - Yi + 0(&4) Y i - &2(1 + ti /12)

Truncating the remainder term yields an implicit three-point fourth-order centereddifference approximation for Y"li:

Consider the second-order boundary-value ODE:

Y" + P(x, y)y' + Q(x, y)y = F(x)

(8.127)

(8.128) Substituting Eqs. (8.115) and (8.127) into Eq. (8.128) yields the implicit fourth-order finite difference equation:

-&-2-(1-+-'--'-J"2/-12-) '2 &(1 + J2/6) ,Y, , (8.129)

If P and/or Q depend on Y, the system of FDEs is nonlinear. If not, the system is linear.