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- verify that the given function is a particular solu-tion to the specified nonhomogeneous equation. Find the general solu-tion and evaluate its arbitrary constants to find the unique solution satisfying the equation and the given initial conditions. y′′ - 2y′ + y = 2ex, yp = x2ex, y(0) = 1, y′(0) =0verify that the given function is a particular solu-tion to the specified nonhomogeneous equation. Find the general solu-tion and evaluate its arbitrary constants to find the unique solution satisfying the equation and the given initial conditions. y′′ - 2y′ + y = x^-1e^x, x >0, yp = xe^x ln x, y(1) = e, y′(1) = 0Find the Fourier coefficients for the function of period 2π as follows: f(x)=2x, -π<x<π Thus obtain its Fourier series expansion.
- Differentiate y=excos(x2): y'= A. excos(x2)-2xsin(x2) B. -2xexsin(x2) C. ex(cos(x2)-2xsin(x2)) D. -exsin(x2)The function f(x) = x is defined on the interval (-pi, pi) and its extension to R is considered as a function of period 2pi. Find its Fourier series.Let f (x) = x sin x and g(x) = x cos x. (a) Show that f ,(x) = g(x) + sin x and g ,(x) = −f (x) + cos x. (b) Verify that f ,,(x) = −f (x) + 2 cos x and g ,,(x) = −g(x) − 2 sin x. (c) By further experimentation, try to find formulas for all higher derivatives of f and g. Hint: The kth derivative depends on whether k = 4n, 4n + 1, 4n + 2, or 4n + 3.