The difference between the two methods in series solutions of ODEs special functions and their need. The power series method is the standard method for using variable coefficients to solve linear ODE's. It's providing solutions in power series form. These series can be used to compute values, to graph curves, to prove formulas and to investigate the properties of solutions. A power series is an infinite series of the form ∑ m = 0 ∞ a m ( x − x 0 ) m = a 0 + a 1 ( x − x 0 ) + a 2 ( x − x 0 ) 2 + ⋅ ⋅ ⋅ here x is a variable, a 0 , a 1 and a 2 ⋅ ⋅ ⋅ are constant called the center of the series. If x 0 = 0 a power series in powers of x as shown below: ∑ m = 0 ∞ a m x m = a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ Power series usually refers to a series of form ∑ m = 0 ∞ a m ( x − x 0 ) m = a 0 + a 1 ( x − x 0 ) + a 2 ( x − x 0 ) 2 + ⋅ ⋅ ⋅ but does not include series of negative or fractional powers of x. The Frobenius method is a generalization of the power series method. It gives solutions in power series, multiplied by a logarithmic term ln x or a fractional power x r . An extended power series refers to a series of form x m ∑ n = 0 ∞ a n x n = x m ( a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ ) . Frobenius method based on solutions have three cases is described below: Let r 1 and r 2 be the roots of the indicial equation. Then the following three cases as: Case 1: Distinct roots not differing by an integer. A basis is y 1 ( x ) = x r 1 ( a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ ) and y 2 ( x ) = x r 2 ( A 0 + A 1 x + A 2 x 2 + ⋅ ⋅ ⋅ ) with coefficients obtained successively with r = r 1 and r = r 2 respectively. Case 2: Double roots r 1 = r 2 = r . A basis is y 1 ( x ) = x r 1 ( a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ ) and y 2 ( x ) = y 1 ( x ) ln x + x r ( A 1 x + A 2 x 2 + ⋅ ⋅ ⋅ ) . Case 3: Roots differing by an integer. A basis is y 1 ( x ) = x r 1 ( a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ ) and y 2 ( x ) = k y 1 ( x ) ln x + x r 2 ( A 0 + A 1 x + A 2 x 2 + ⋅ ⋅ ⋅ ) where the roots are so denoted that r 1 − r 2 > 0 and k may turn out to be zero.

Question

No chatgpt pls will upvote

Answer 1

Expert Solution & Answer

Trending nowThis is a popular solution!

See solution

Check out sample textbook solution

Students have asked these similar questions

No chatgpt pls will upvote

3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.

matrix 4