4) The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. y = c1e^4x + c2e^-x, (-?, ?); y'' - 3y' - 4y = 0, y(0) = 1, y'(0) = 1 5) The indicated function y1(x) is a solution of the given differential equation. x^2y'' - xy' + 5y = 0; y1 = x cos(2 ln(x)) Use reduction of order or formula to find a second solution y2(x).
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So, the general solution is \(y = c_1e^{4x} + c_2e^{-x}\). Show more…
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In Problems 1–4 the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. 1. y = c1 e^x + c2 e^{-x}, (-∞, ∞); y'' - y = 0, y(0) = 0, y'(0) = 1 2. y = c1 e^{4x} + c2 e^{-x}, (-∞, ∞); y'' - 3y' - 4y = 0, y(0) = 1, y'(0) = 2 3. y = c1 x + c2 x ln x, (0, ∞); x^2 y'' - x y' + y = 0, y(1) = 3, y'(1) = -1 4. y = c1 + c2 cos x + c3 sin x, (-∞, ∞); y'' + y' = 0, y(π) = 0, y'(π) = 2, y''(π) = -1
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The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
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