5. Find two power series solutions of the DE about the ordinary point x = 0: y"+y = 0 Give at least 4 terms of each series.

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A: See the detailed solution below.

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Q: 4) Find a power series for f(x) = X + cos vx and and state its interval of convergence

A: The power series of cos x is: cos x=∑n=1∞(-1)nx2n(2n)! Replace x by x: cos x=∑n=0∞(-1)nx2n(2n)!…

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A: Let the solution of the equation be y=∑n=1∞antn. On differentiating, y'=∑nn=1∞antn-1 Again…

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A: The integral is ∫1+x3 dx. The series expansion of 1+xk is 1+xk=∑n=0∞knxn. Here, the function is…

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A: The given differential equation is y"+y'+x2y=0

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A: We use the power series method to find the solution of the given differential equation.

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Q: 5. Find two linearly independent power series solutions around xo = 0 to y" – xy = 0. %3D

A: Given below the detailed solution.

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A: Given : It represent Geometric series :

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A: Given differential equation is y''+x2y = 0.

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A: ∑k=1∞ 3x+5k2k

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A: Given for a complex power series is the limit L=3|z-i|.

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A: The expansion for ln1+x can be written as follows:

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A: Given: xy''+2y'-xy=sech2(y) To find: Power series

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