Variation of Parameters
Power Series Method
CS 130: Mathematical Methods in Computer Science Ordinary Differential Equations Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 16
Variation of Parameters
Power Series Method
Ordinary Differential Equations
Variation of Parameters
Power Series Method
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x.
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x. → yc :
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x. → yc : aux eqn : m2 + 1 = 0
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x. → yc : aux eqn : m2 + 1 = 0 roots: m = ±i
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x. → yc : aux eqn : m2 + 1 = 0 roots: m = ±i yc = c1 cos x + c2 sin x
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x. → yc : aux eqn : m2 + 1 = 0 roots: m = ±i yc = c1 cos x + c2 sin x → yp :
Variation of Parameters
Power Series Method
Variation of Parameters using Wronskian If the nonhomogeneous equation y 00 + P (x)y 0 + Q(x)y = f (x) has complementary function yc = c1 y1 (x) + c2 y2 (x), then a particular solution is given by Z Z y1 (x)f (x) y2 (x)f (x) dx + y2 (x) dx, yp = −y1 (x) W (x) W (x) where W (x) is the Wronskian of the two independent solutions y1 and y2 of the associated homogeneous equation.
Example Solve y 00 + y = tan x. → yc : aux eqn : m2 + 1 = 0 roots: m = ±i yc = c1 cos x + c2 sin x → yp :
Variation of Parameters
Power Series Method
Power Series
Recall: A power series in x − a is an infinite series of the form ∞ X
cn (x − a)n = c0 + c1 (x − a) + c2 (x − a)2 + · · · + cn (x − a)n + · · ·
n=0
This power series converges on the interval I provided that the limit ∞ X n=0
cn (x − a)n = lim
N →∞
N X
cn (x − a)n
n=0
exists for all x in I. In this case, f (x) =
∞ P
cn (x − a)n is defined on
n=0
I, and we call the series a power series representation of f on I.
Variation of Parameters
Power Series Method
Power Series Some familiar power series representations from introductory calculus:
Variation of Parameters
Power Series Method
Power Series Method
The power series method for solving a differential equation consists of substituting the power series y=
∞ X
cn xn
n=0
in the differential equation and then attempting to determine what the coefficients c0 , c1 , c2 , · · · must be in order that the power series will satisfy the differential equation.
Variation of Parameters
Power Series Method
Termwise Differentiation of Power Series
If the power series representation f (x) =
∞ X
cn xn
n=0
of the function f converges on the open interval I, then f is differentiable on I, and f 0 (x) =
∞ X n=1
at each point on I.
ncn xn−1
Variation of Parameters
Power Series Method
Identity Principle
If
∞ P
an xn =
n=0 a n = bn
∞ P
bn xn for every point x in some open interval I, then
n=0
for all n ≥ 0.
In particular, if
∞ P
an xn = 0 for all x in some open interval, it follows
n=0
that an = 0 for all n ≥ 0.
Variation of Parameters
Power Series Method
Example Solve the equation y 0 + 2y = 0. Substitute the series y =
∞ P n=0
cn xn and obtain
Variation of Parameters
Power Series Method
Example Solve the equation y 0 + 2y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 1)cn+1 + 2cn ]xn = 0
n=0
Variation of Parameters
Power Series Method
Example Solve the equation y 0 + 2y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 1)cn+1 + 2cn ]xn = 0
n=0
2cn We get the recurrence relation cn+1 = − n+1 for all n ≥ 0.
Variation of Parameters
Power Series Method
Example Solve the equation y 0 + 2y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 1)cn+1 + 2cn ]xn = 0
n=0
2cn We get the recurrence relation cn+1 = − n+1 for all n ≥ 0.
We can easily prove by induction on n that 2n c0 cn = (−1)n , n ≥ 1. n!
Variation of Parameters
Power Series Method
Example Solve the equation y 0 + 2y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 1)cn+1 + 2cn ]xn = 0
n=0
2cn We get the recurrence relation cn+1 = − n+1 for all n ≥ 0.
We can easily prove by induction on n that 2n c0 cn = (−1)n , n ≥ 1. n! Consequently, our solution takes the form ∞ ∞ ∞ X X X 2n c0 n (−2x)n y= cn xn = (−1)n x = c0 = c0 e−2x n! n! n=0 n=0 n=0
Variation of Parameters
Power Series Method
Example Solve the equation y 00 + y = 0. Substitute the series y =
∞ P n=0
cn xn and obtain
Variation of Parameters
Power Series Method
Example Solve the equation y 00 + y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 2)(n + 1)cn+2 + cn ]xn = 0
n=0
Variation of Parameters
Power Series Method
Example Solve the equation y 00 + y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 2)(n + 1)cn+2 + cn ]xn = 0
n=0
cn We get the recurrence relation cn+2 = − (n+1)(n+2) for all n ≥ 0.
Variation of Parameters
Power Series Method
Example Solve the equation y 00 + y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 2)(n + 1)cn+2 + cn ]xn = 0
n=0
cn We get the recurrence relation cn+2 = − (n+1)(n+2) for all n ≥ 0.
Prove by induction that for k ≥ 1, c2k =
(−1)k c0 (−1)k c1 and c2k+1 = . (2k)! (2k + 1)!
Variation of Parameters
Power Series Method
Example Solve the equation y 00 + y = 0. Substitute the series y =
∞ P
cn xn and obtain
n=0 ∞ X
[(n + 2)(n + 1)cn+2 + cn ]xn = 0
n=0
cn We get the recurrence relation cn+2 = − (n+1)(n+2) for all n ≥ 0.
Prove by induction that for k ≥ 1, c2k =
(−1)k c0 (−1)k c1 and c2k+1 = . (2k)! (2k + 1)!
Consequently, our solution takes the form y = c0 cos x + c1 sin x
Variation of Parameters
Power Series Method
Questions? See you next meeting!