Q3 Consider the following second-order ODE: (2 – x)xy" (x) + (x² – 2)y'(x) – 2(x – 1)y(x) = -x². Find a polynomial solution of the respective homogeneous ODE. In other words, you want to look for a function y(x) = ao + ..· + anxWhich solves the homogeneous ODE. Hint: I can guarantee that there is a solution which is a polynomial of degree < 2. Find the general solution of the homogeneous ODE. You are allowed to use the Abel's formula. 'Find the general solution of the given non-homogeneous ODE. You are allowed to take advantage of the fact that there is a particular solution which is also a polyniomal. You can still use the variation of parameters, if you want.

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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.