Word Count : 666

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We are given the second-order linear ordinary differential equation (ODE): \[ \frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = -6\sin(2x) - 18\cos(2x) \] with the initial conditions: \[ y(0) = 2, \quad y'(0) = 2. \] ### Step 1: Solve the homogeneous equation The homogeneous part of the equation is: \[ \frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = 0. \] The characteristic equation corresponding to this is: \[ r^2 + r - 2 = 0. \] We can solve this quadratic equation using the quadratic formula: \[ r = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm \sqrt{9}}{2} = \frac{-1 \pm 3}{2}. \] Thus, the two roots are: \[ r_1 = 1, \quad r_2 = -2. \] So the general solution to the homogeneous equation is: \[ y_h(x) = C_1 __________ _______ ___ ________ ____ ______ ________ __________ ______ _______ ______.
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