1. (20pts) (a) Find general solutions to the following differential equation $4y'' - 4y' + y = 0$. (b) Find the unique solution which satisfies initial condition $y(0) = 1$ and $y'(0) = -2$. 2. (20pts) If the method of undetermined coefficients is used, write down (but do NOT solve) the correct form for a particular solution $y_p$ to the equation $y'' + 3y' + 2y = f(x)$ when $f(x)$ equals (a) $5e^{-2x}$ (b) $x^2 cos 9x$ (c) $2xe^{-x} sin x - e^{-x} cos x$ (d) $7 cos 3x$ 3. (15pts) Find general solutions to the following non-homogeneous equation: $y'' - y' - 12y = 2 sin x$
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First, we need to find the general solution to the given differential equation: 4y'' - 4y' + y = 0 This is a linear homogeneous differential equation with constant coefficients. We can solve it by assuming a solution of the form y = e^(rx), where r is a Show more…
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1. Use the Method of Undetermined Coefficients to give the general solution of the following nonhomogeneous equation. y'' - 3y' - 4y = 2 sin t 2. Find the general solution to the differential equation. As you are working through this problem, use the Method of Undetermined Coefficients to find the particular solution. y'' + y' - 12y = e^t + e^{2t} - 1 3. Use the Method of Undetermined Coefficients to give the general solution of the following nonhomogeneous equation. d^2y/dx^2 - 5dy/dx + 6y = xe^x 4. Use the Method of Undetermined Coefficients to give the general solution of the following nonhomogeneous equation. y'' - 3y' + 2y = e^x sin x HINT and TIMESAVER: You may use the fact that the general solution to the corresponding homogeneous equation is yh(x) = c1e^x + c2e^{2x}.
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