Explanation Given information: The given differential equation is y ' + k y = 0 . Formula used: Power series is defined as an infinite series of the form, ∑ n = 0 ∞ a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ......... + a n x n + .... , where x is a variable. Proof: Assume y = ∑ n = 0 ∞ a n x n is a solution of the differential equation. Then, differentiating the assumed solution with respect to x , y ' = ∑ n = 1 ∞ n a n x n − 1 Now, the given differential equation is, y ' + k y = 0 Inserting the value of y and y ' in above equation, ∑ n = 1 ∞ n a n x n − 1 + k ∑ n = 0 ∞ a n x n = 0 ∑ n = 1 ∞ n a n x n − 1 = − k ∑ n = 0 ∞ a n x n To obtain equal power of x , adjust the summation indices by replacing n by n +1 in the left hand side sum to obtain, ∑ n = 0 ∞ ( n + 1 ) a n + 1 x n = − k ∑ n = 0 ∞ a n x n Equating the coefficients of like terms, one can obtain the recursion formula

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ISBN: 9781337275378

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 16.4, Problem 16E

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To verify: That the solution of the differential equation y'+ky=0 is same as obtained by using a power series and by any other method.

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Chapter 16.4, Problem 15E

Chapter 16.4, Problem 17E

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Solve the differential Equation: NONHOMOGENEOUS EQUATION: The Method of Undetermined Coefficients (show complete and neat solution)

Tutorial Exercise Consider the differential equation x²y" - 4xy' + 6y=0; x², x³², (0, on). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy + 6y=0; x², x³, (0, 0) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c, and c₂, not both zero, s are different powers of x, we have a formal test to verify the linear independence. Recall the definition of the Wronskian for the case of two functions f, and f, each of which have a first derivative. f₂ W(f₁, f₂) = Will HOUTBEave any pronTIES FOR LA By Theorem 4.1.3, if W(f₁, f₂) 0 for every x in the interval of the solution, then solutions are linearly independent. Let f(x) = x² and f(x) = x³. Complete the Wronskian for these functions. x² w(x², x³) = 2x

Solve the differential equation by using power series method? y'-xy=0

Chapter 16 Solutions

Multivariable Calculus

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To verify: That the solution of the differential equation y ' + k y = 0 is same as obtained by using a power series and by any other method.

Solution Summary: The author explains that the solution of the differential equation y'+ky=0 is the same as obtained by using a power series.

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To verify: That the solution of the differential equation y'+ky=0 is same as obtained by using a power series and by any other method.

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Chapter 16 Solutions