Part A) Use the Taylor series expansion of sin x to write sin(√(t)) = t^(1/2) - (t^3/2)/3! + (t^5/2)/5! - ...   Part B) Solve the following first order equation (using Laplace transform or power series method) y'+y= sin(√(t)), y(0)=1.

Q: Find the formal Fourier series solutions of the endpoint value problem. x'' + 7x = 5t, x(0) = x(1)…

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Q: Calculate the value of the function f(x) = sinx cosx at  x= 0.1 using the 4th order derivative…

A: Suppose fx is n times differentiable function then nth degree Taylor approximation for fx near 0 as…

Q: (a) Solution. Compute the Fourier sine series of the function f(x) = x on the interval [0, 7]. We…

A: Given function is  The Fourier  sine  series  of the  function  is  given by

Q: Consider the function f(x) = (8)3. We want to find the Taylor series of f(x) at x=7 and at x = -5.…

A: ### (a) Finding the nth Derivative of  f(x)The function is f(x)=(8−x)32​. To find the nth…

Q: The Fourier sine series of the function f(x) = x³ on the interval [0, π] is The solution of the…

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Q: Find the solution to the ODE using power series solutions   (x-1)y''+3y=0

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Q: find the first few terms of the taylor series of f(x) = 1/x at a = 1, solve by replacing x with…

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Q: 5. (a)Find the Fourier sine series for f(x) = x³ – n²x on the interval [-7, T]. (b) By plugging in a…

A: For (a), We need to find the Fourier sine series for the function, f(x) = x3 - π2x on the interval…

Q: Problem 7. Find a periodic Solution as a Fourier series to x" + 3x = F(t) Where F(t) = 2t on…

A: (a) Given that F(t)=2tF(−t)=−2t=−F(t)Therefore F(t) is an odd function.F(t) extends to [−π,π] so the…

Q: Solve the ODE using power series y'=y+1

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Q: Solve the ODE using Power Series y'' + x^2 y=0,  y(0)=0

A: We have to solve given ODE:

Q: expand the function f(x) = sin(x) / (coshx +2) in a taylor series around the origin going up to x^3.…

A: Taylor series of function f(x) around x=a is defined as:

Q: Consider the 27-periodic function f(x) defined on [-, π] by the formula f(x) = cos(). (a) Find the…

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Q: f(x) 0 if if 0 -2 < x < 0 Find the Fourier series for the function 3x Sx <2 Expand and write out the…

A: For the given function take l = 4.   The Fourier series for the function is given as:

Q: After separation of variables, if the constant is a negative value, how is the Fourier series…

A: Given  the one dimensional heat equation with some boundary conditions, i.e  ∂u∂t=k∂2u∂x2…