First-Order Equations. The series methods discussed in this section are directly applicable to the first-order linear differential equation P(x)y′ + Q(x)y = 0 at a point x0, if the function p = Q/P has a Taylor series expansion about that point. Such a point is called an ordinary point, and further, the radius of convergence of the series y=∞∑n=0an(x−x0)ny=∑n=0∞anx−x0n is at least as large as the radius of convergence of the series for Q/P. In each of Problems 13 through 16, solve the given differential equation by a series in powers of x and verify that a0 is arbitrary in each case. Problem 17 involves a nonhomogeneous differential equation to which series methods can be easily extended. Where possible, compare the series solution with the solution obtained by using the methods of Chapter 2. 14.y′ − xy = 0
First-Order Equations. The series methods discussed in this section are directly applicable to the first-order linear differential equation P(x)y′ + Q(x)y = 0 at a point x0, if the function p = Q/P has a Taylor series expansion about that point. Such a point is called an ordinary point, and further, the radius of convergence of the series y=∞∑n=0an(x−x0)ny=∑n=0∞anx−x0n is at least as large as the radius of convergence of the series for Q/P. In each of Problems 13 through 16, solve the given differential equation by a series in powers of x and verify that a0 is arbitrary in each case. Problem 17 involves a nonhomogeneous differential equation to which series methods can be easily extended. Where possible, compare the series solution with the solution obtained by using the methods of Chapter 2.
14.y′ − xy = 0
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