We consider the non-homogeneous problem y" - 4y = 16x First we consider the homogeneous problem y"- 4y = 0: 1) the auxiliary equation is ar? + br + c = ^2-4 0. 2) The roots of the auxiliary equation are 2,-2 (enter answers as a comma separated list). 3) A fundamental set of solutions is e^(2x),e^(-2x) (enter answers as a comma separated list). Using these we obtain the the complementary solution Ye = C1y1 + cC2y2 for arbitrary constants c and c2. Next we seek a particular solution y, of the non-homogeneous problem y" - 4y = 16x using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find y, We then find the general solution as a sum of the complementary solution Ye = Ciy1 + C2Y2 and a particular solution: y = Ye + Yp• Finally you are asked to %3D use the general solution to solve an IVP. 5) Given the initial conditions y(0) = -2 and y' (0) = -14 find the unique solution to %3D %3D the IVP y =

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.