220 Ordinary Differential Equationswhere an, an_ 1 , ... , a 1 , and ao are constants and where an -=/=- 0 subject to
the initial conditionsThis initial value problem has a unique solution on the largest interval con-
taining x 0 on which the function b(x) is continuous. When b(x) is a linear
combination of terms of the form (6), (7), (8), and (9) of section 4.4, the
solution of the initial value problem (3)-(4) can be found by calculating the
complementary solution, Ye, which will include n arbitrary constants c1, c2,
... , en; by calculating a particular solution, yP, using the method of undeter-
mined coefficients; and then by determining values for the constants c 1 , c2,
... , Cn such that y = Ye + Yp satisfies the equ ations of ( 4 ) -this requires the
solution of a system of n linear equations inn unknowns. In this instance, you
must remember that it is the coefficients of the general so lution y = Ye +yp and
not the coefficients of the complementary solution Ye which must be chosen
to satisfy the initial conditions (4).
A second method for solving the initial value problems (1)-(2) and (3)-
(4) as well as the general initial value problem consisting of the differential
equation y(n)(x) = J(x, y(x), yC^1 l(x), ... , y(n-l)(x)) and the initial conditionsy(xo) = k1, yC^1 l(xo) = k2, ... , y(n-l)(xo) = kn is to rewrite these n-th
order differential equations as a system of n first-order differential equations
and then use the techniques described in chapter 7 to solve the corresponding
system initial value problem. When an explicit equation for the solution is
not required, the second method of solution is the simplest method to use.
This method will produce the solution of the initial value problem as a set
of ordered pairs (a function) on any finite interval about x 0 in which certain
conditions on f, fy, fy<'l, ... , fy<n-1) are satisfied.EXAMPLE 1 Solution of a Nonhomogeneous
Initial Value ProblemSolve the initial value problem(5) y" + 3y' = 4x^2 - 2e-^3 "'; y(O) = 1, y' (0) = 0.