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

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Q1 Consider The Third Order Linear Homogeneous Ordinary Differential Equa Tion With Variable Coefficients Dy 2 X 1 (57.1 KiB) Viewed 26 times

Q1. Consider the third-order linear homogeneous ordinary differential equa- tion with variable coefficients dy (2 - x) + (2x - 3) +y = 0, < 2. dc d'y day C d.x3 d. 2 - 2 First, given that yı(2) = e" 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(x) must be explicitly determined. Now, consider the inhomogeneous ordinary differential equation day dy (2-x) + y = (x - 2)2, x < 2. < 2 dac Let y(x) = u1(x)f(x) + u2(r)g(x) + 43(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) dr3 + (2x - 3) d.x2