SERIES SOLUTION OF DIFFERENTIAL EQUATIONS
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1 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS Introduction to Differential Equations Nanang Susyanto Computer Science (International) FMIPA UGM 17 April 2017 NS (CS-International) Series solution 17/04/ / 11
2 What you have to recall Derivative Taylor Series (Exponential, Trigonometric, Logarithmic, etc.) NS (CS-International) Series solution 17/04/ / 11
3 Analytic Functions A function f (x) of one variable x is said to be analytic at a point x = x 0 if it has a convergent power series expansion f (x) = 0 a n (x x 0 ) n = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) a n (x x 0 ) n + for x x 0 < R, R > 0. This point x = x 0 is also called ordinary point. NS (CS-International) Series solution 17/04/ / 11
4 Analytic Functions A function f (x) of one variable x is said to be analytic at a point x = x 0 if it has a convergent power series expansion f (x) = 0 a n (x x 0 ) n = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) a n (x x 0 ) n + for x x 0 < R, R > 0. This point x = x 0 is also called ordinary point. Otherwise, f (x) is said to have a singularity at x = x 0. NS (CS-International) Series solution 17/04/ / 11
5 Analytic Functions A function f (x) of one variable x is said to be analytic at a point x = x 0 if it has a convergent power series expansion f (x) = 0 a n (x x 0 ) n = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) a n (x x 0 ) n + for x x 0 < R, R > 0. This point x = x 0 is also called ordinary point. Otherwise, f (x) is said to have a singularity at x = x 0. The largest such R (possibly +) is called the radius of convergence of the power series. NS (CS-International) Series solution 17/04/ / 11
6 Radius Convergence Radius convergence: The largest such R (possibly +) such that the series converges for every x with x x 0 < R and diverges for every x with x x 0 > R. There is a formula for R = 1 l, where l = if the latter limit exists. 1, or lim lim n a n 1/n n a n+1, a n NS (CS-International) Series solution 17/04/ / 11
7 Properties of Analytic Functions If f (x), g(x) are analytic at x = x 0 then so is f (x)g(x) and af + bg for any scalars a, b with radii of convergence at least that of the smaller of the radii of convergence the series for f (x), g(x). NS (CS-International) Series solution 17/04/ / 11
8 Properties of Analytic Functions If f (x), g(x) are analytic at x = x 0 then so is f (x)g(x) and af + bg for any scalars a, b with radii of convergence at least that of the smaller of the radii of convergence the series for f (x), g(x). If f (x) is analytic at x = x 0 and f (x 0 ) = 0 then 1/f (x 0 ) is analytic at x = x 0. NS (CS-International) Series solution 17/04/ / 11
9 Properties of Analytic Functions If f (x), g(x) are analytic at x = x 0 then so is f (x)g(x) and af + bg for any scalars a, b with radii of convergence at least that of the smaller of the radii of convergence the series for f (x), g(x). If f (x) is analytic at x = x 0 and f (x 0 ) = 0 then 1/f (x 0 ) is analytic at x = x 0. Power series can be integrated and differentiated within the interval (disk) of convergence i.e., for x x 0 < R we have f (x) = na n x n 1 = n=1 n=0 (n + 1)a n+1 x n, and the resulting power series have R as radius of convergence. NS (CS-International) Series solution 17/04/ / 11
10 Series Solutions near a Ordinary Point NS (CS-International) Series solution 17/04/ / 11
11 Series Solutions near a Ordinary Point Theorem If p 1 (x), p 2 (x),..., p n (x), q(x) are analytic at x = x 0, the solutions of the DE y (n) + p 1 (x)y (n 1) + + p n (x)y = q(x) are analytic with radius of convergence the smallest of the radii of convergence of the coefficient functions p 1 (x), p 2 (x),..., p n (x), q(x). NS (CS-International) Series solution 17/04/ / 11
12 Series Solutions near a Ordinary Point Theorem If p 1 (x), p 2 (x),..., p n (x), q(x) are analytic at x = x 0, the solutions of the DE y (n) + p 1 (x)y (n 1) + + p n (x)y = q(x) are analytic with radius of convergence the smallest of the radii of convergence of the coefficient functions p 1 (x), p 2 (x),..., p n (x), q(x). The proof of this result follows from the proof of fundamental existence and uniqueness theorem for linear DE s using elementary properties of analytic functions and the fact that uniform limits of analytic functions are analytic. NS (CS-International) Series solution 17/04/ / 11
13 Easy Example Solve the DE by series. y = y NS (CS-International) Series solution 17/04/ / 11
14 Example 1 (page 1 from 3) Solve the DE y + y = 0. NS (CS-International) Series solution 17/04/ / 11
15 Example 1 (page 1 from 3) Solve the DE y + y = 0. Answer: The coefficients of the DE y + y = 0 are analytic everywhere, in particular at x = 0. Any solution y = y(x) has therefore a series representation y = a n x n n=0 with infinite radius of convergence. We have y = n=0 (n + 1)a n+1 x n, y = n=0 (n + 1)(n + 2)a n+2 x n. NS (CS-International) Series solution 17/04/ / 11
16 Example 1 (page 2 from 3) Therefore, we have y + y = n=0 ((n + 1)(n + 2)a n+2 + a n )x n = 0 for all x. It follows that (n + 1)(n + 2)a n+2 + a n = 0 for n 0. Thus a n a n+2 = (n + 1)(n + 2), for n 0 from which we obtain a 2 = a 0 1 2, a 3 = a 1 2 3, a 4 = a = a , a 5 = a = a NS (CS-International) Series solution 17/04/ / 11
17 Example 1 (page 3 from 3) By induction one obtains and hence that a 2n = ( 1) n a 0 (2n)!, a 2n+1 = ( 1) n a 1 (2n + 1)! x 2n y = a 0 ( 1) n (2n)! + a 1 n=0 = a 0 cos(x) + a 1 sin(x). n=0 ( 1) n x 2n+1 (2n + 1)! NS (CS-International) Series solution 17/04/ / 11
18 Airy s equation Solve the Airy s equation y + xy = 0 by series. NS (CS-International) Series solution 17/04/ / 11
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