Q1. Consider the third-order linear homogeneous ordinary differential equa- tion with variable coefficients dy dạy (2-x)

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Q1. Consider the third-order linear homogeneous ordinary differential equa- tion with variable coefficients dy dạy (2-x) + (2x - 3) +y=0, < 2. d.x2 dc dy d.r3 First, given that y(x) = er is a solution of the above equation, use the method of reduction of order to find its general solution as yn(x) = Cif(x) + C29(x) + C3h(x), where the functions f(x), g(x), h(c) must be explicitly determined. Now, consider the inhomogeneous ordinary differential equation day dy (2-2) + (2x - 3) << 2. + y = (x - 2)2, dr2 dx Let y(x) = u1(x)f(x) + u2(x)g(x) + u3(x)h(x) and use the method of variation of parameters to write down the three ordinary differential equations that must be satisfied by the first-order derivatives of the unknown functions ui, U2, U3. Find these functions by integration, and thus establish the particular solution yp(x) of the given inhomogeneous equation. (30 marks) 0.23 -