The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation
y'' + p(t)y' + q(t)y = g(t), (38)
provided one solution y1 of the corresponding homogeneous equation is known. Let y = v(t)y1(t) and show that y satisfies equation 38 if v is a solution of
y1(t)v'' + (2y1'(t) + p(t)y1(t))v' = g(t). (39)
Equation 39 is a first order linear equation for v'. Solving this equation, integrating the result, and then multiplying by y1(t) leads to the general solution of the first equation.
Use the method above to solve the differential equation
t^2y'' - 2ty' + 2y = 4t^2, t > 0, y1(t) = t.
NOTE: Use C1, C2, ... for the constants of integration.
Y(t) =