Final answer:
The student inquires about solving various second-order differential equations, which require finding general solutions for the homogeneous part and particular solutions for the non-homogeneous part.
Step-by-step explanation:
The student is asking how to solve various second-order differential equations. These equations are more complex than first-order because they involve the second derivative of the function which can represent acceleration or curvature depending on the context. To solve these, one must typically find a particular solution that satisfies the non-homogeneous part and a general solution that satisfies the homogeneous equation.
For equation (a), y'' - 2y' + 2y = 0, we seek a solution such that y(π) = 0 and y(-π) = 0. This involves finding the characteristic equation and its roots, and then applying boundary conditions to find the particular constants for the general solution.
Equation (b), x^2 y'' - 10xy' + 18y = 0, is a Cauchy-Euler equation which is solved by assuming a solution of the form y = x^m and solving for m to obtain the general solution.
Equation (c), y'' - y - 6 = x^2 + 1, is a non-homogeneous linear equation. Here, one typically finds the general solution to the complementary homogeneous equation y'' - y - 6 = 0 and a particular solution to the non-homogeneous equation using methods like undetermined coefficients or variation of parameters.
For equation (d), y'' + 2y' + y = xe^{-x}, similar to the previous equation, it involves a non-homogeneous linear differential equation where the right side is the non-homogeneous part and requires a method like variation of parameters to find a particular solution, along with the general solution of the homogeneous part.