(20 pts) Find a general solution to y" - 2y' + y = 2et/(1 + t2).
Finding a solution takes several steps that I'll combine into two.
Find and solve the characteristic equation. r2 - 2r + 1 = 0 has two roots both equal to 1. So we have two independent solutions to the homogeneous equation
Use Variation of Parameters to find a particular solution to the nonhomogeneous equation
Let g(t) = 2et/(1 + t2). Then
where u1(t) = - Integral(y2g·dt/W(y1, y2)), while u2(t) = Integral(y1g·dt/W(y1, y2)). The Wronskian
Therefore we can take
Therefore we can take u2(t) = 2tan-1(t).
We now have a general solution: