We solve the following initial-value problems using twodifferent methods (Runge Kutta – 4th Order Taylor series –Finite difference method (Three choose two of the methods.)))and solve it. x E [0, 20] for x-y draw the graphs. (h to bedetermined by you.)a) y"+0.5(y² – 1)y'+ y=0, y(0) = 1, y'(0) = 0b) y"=y cos(2x), y(0) = 0, y'(0) =1

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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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Let f(x)=e−xf(x)=e−x. We're going to calculate the Taylor series for this function near a=0a=0 . Calculate f(0),f′(0),f′′(0),f(0),f′(0),f″(0), and f(3)(0)f(3)(0) . What patterns do you notice here? Write down the degree 3 Taylor polynomial approximation for f(x)f(x) near a=0a=0 . Make a graph that shows the function and the degree 3 Taylor polynomial. For what interval of x-values is this a relatively good approximation? For which x-values is this not a very good approximation? Based on the patterns you observed in (a), find a general formula for the k-th derivative of this function when evaluated at 0.f(k)(0)=?f(k)(0)=? Write the Taylor series fo f(x)f(x) near a=0a=0, using sigma notation. What is the interval of convergence for this series? What would be different about your result if we instead calculated the Taylor series for f(x)f(x) near a=4a=4 ?
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We solve the following initial-value problems using two different methods (Runge Kutta – 4th Order Taylor series Finite difference method (Three choose two of the methods.))) and solve it. x E [0, 20] for x-y draw the graphs. (h to be determined by you.) -