Consider the following first order differential equation. Let \( y = \sum_{n=0}^{\infty} a_n x^n \) be a power series solution in \( x \) to the equation.\[ y' - 4y = 0, \quad y(0) = 3 \]a) Find the general coefficient \( a_n \) of the power series of the solution \( y \) in terms of \( n \).\[ a_n = \underline{\hspace{3cm}} \]b) Calculate the radius of convergence (use \( \infty \) if the radius is infinite).\[ R = \underline{\hspace{3cm}} \]c) Express the solution in terms of familiar elementary functions.\[ y(x) = \underline{\hspace{3cm}} \]

Let us solve above differential equation.

Consider the following first order differential equation. Let y = Σanx" be a power series solution in a n=0 to the equation. y'-4y= 0, y(0) = 3 a) Find the general coefficient an of the power series of the solution y in terms of n. an b) Calculate the radius of convergence (use oo if the radius is infinite). R= ∞ c) Express the solution in terms of familiar elementary functions. y(x) = =

Consider the following first order differential equation. Let y = Σanx" be a power series solution in a n=0 to the equation. y'-4y= 0, y(0) = 3 a) Find the general coefficient an of the power series of the solution y in terms of n. an b) Calculate the radius of convergence (use oo if the radius is infinite). R= ∞ c) Express the solution in terms of familiar elementary functions. y(x) = =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the following first order differential equation. Let \( y = \sum_{n=0}^{\infty} a_n x^n \) be a power series solution in \( x \) to the equation.

\[ y' - 4y = 0, \quad y(0) = 3 \]

a) Find the general coefficient \( a_n \) of the power series of the solution \( y \) in terms of \( n \).

\[ a_n = \underline{\hspace{3cm}} \]

b) Calculate the radius of convergence (use \( \infty \) if the radius is infinite).

\[ R = \underline{\hspace{3cm}} \]

c) Express the solution in terms of familiar elementary functions.

\[ y(x) = \underline{\hspace{3cm}} \]

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