Solving $t^2y''-t(t+2)y'+(t+2)y=2t^3$ knowing $y_1 = t, y_2 = te^t$ are solutions to the homogeneous equation $$t^2y''-t(t+2)y'+(t+2)y=2t^3\implies\\y''-\frac{t+2}{t}+\frac{t+2}{t^2}y = 2t$$

My book says that $y_1 = t, y_2 = te^t$ are solutions to the homogeneous equation.

I'm trying a solution of the form $Y = u_1t+u_2te^t$, so the wronskian is

$$W(t, te^t) = t^2e^t$$

Then, the solutions are given by:

$$u_1 = \int\frac{-te^t\cdot 2t}{t^2e^t} \ dt= -2t$$
$$u_2 =  \int\frac{t\cdot 2t}{t^2e^t} \ dt=-2e^{-t}$$

Then, the solution will be:

$$Y = u_1y_1+u_2y_2 = -2t\cdot t + -2e^{-t}\cdot te^t = -2t^2-2t$$

but my book gives $-2t^2$ as the answer. What am I doing wrong?
t is belong to the homogenous equation : y = c1t + c2te^t
the book only give the particular solution part. http://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx

maybe... :)
t is belong to the homogenous equation : y = c1t + c2te^t
the book only give the particular solution part. http://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx

maybe... :)