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15. Let f(x) be a function such that f(0) = 2, f'(x) = 3f (x), and the n" derivative of f is given by f(") (x)=3f-1) (x). (a) Give the first four terms and the general term of the Taylor Series for f centered at x = 0. (b) Find f(x) by solving the differential equation f'(x)=3f(x) (that is y'=3y) with the initial condition f(0) = 2.

15. Let f(x) be a function such that f(0) = 2, f'(x) = 3f (x), and the n" derivative of f is given by f(") (x)=3f-1) (x). (a) Give the first four terms and the general term of the Taylor Series for f centered at x = 0. (b) Find f(x) by solving the differential equation f'(x)=3f(x) (that is y'=3y) with the initial condition f(0) = 2.

Advanced Engineering Mathematics
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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Problem number 15- a+b I’m not sure how to find a Taylor series without the root function. I also struggle with derivatives and integrals when y and y’ are involved. Please help:)
J (²) Also give the interval of convergence. 1+x 1 about x =2. 12. Write four terms and the general term of the Taylor series expansion of f(x)=x-1 13. Use the series from Problem 12 to find four terms and the general term of the series expansion about x=2 for In|x-1. 14. The Taylor Series of a function about x=3 is given by f(x)=1+³(x-3) 5(x−3)²_7(x-3)³ (2n+1)(x-3)" n! +... What is the value of f" (3)? 1! 2! 3! 15. Let f(x) be a function such that f(0) = 2, f'(x) = 3f (x), and the n" derivative of f is given by f(") (x)=3f(-¹) (x). + ++ (a) Give the first four terms and the general term of the Taylor Series for f centered at x = 0. (b) Find f(x) by solving the differential equation f'(x)=3f(x) (that is y'=3y) with the initial condition f(0) = 2.
Transcribed Image Text:J (²) Also give the interval of convergence. 1+x 1 about x =2. 12. Write four terms and the general term of the Taylor series expansion of f(x)=x-1 13. Use the series from Problem 12 to find four terms and the general term of the series expansion about x=2 for In|x-1. 14. The Taylor Series of a function about x=3 is given by f(x)=1+³(x-3) 5(x−3)²_7(x-3)³ (2n+1)(x-3)" n! +... What is the value of f" (3)? 1! 2! 3! 15. Let f(x) be a function such that f(0) = 2, f'(x) = 3f (x), and the n" derivative of f is given by f(") (x)=3f(-¹) (x). + ++ (a) Give the first four terms and the general term of the Taylor Series for f centered at x = 0. (b) Find f(x) by solving the differential equation f'(x)=3f(x) (that is y'=3y) with the initial condition f(0) = 2.
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