Approach towards second order differential equation I have the following equation to be solved. Can anybody explain to me how I am supposed to approach this problem?
$$4 \frac{d^{2}y}{dx^{2}} + 4 \frac{dy}{dx} + y = (8x^{2} + 6x + 2)e^{-x/2}$$
edited
I am supposed to find the particular integral for the same.
 A: Hint 1: First find a homogeneous solution for $4y''+4y'+y = 0$.
Hint 2: $4y^2+4y+1 = 0 \to (2y+1)^2 = 0 \to y_h = Ae^{-x/2} + Bxe^{-x/2}, y_p = (Cx^2+Dx+E)e^{-x/2}$
Hint 3: $y = y_h+y_p$
A: First find the general solution of the homogeneous equation $4y''+4y'+y=0$ in the usual way, using the characteristic equation $4r^2+4r+1=0$. Then use the method of undetermined coefficients to find a particular solution of the nonhomogeneous equation. In "guessing" your undetermined coefficients form for the particular solution, be sure to take into account the fact that some terms of the input function (forcing function) $(8x^2+6x+2)e^{-x/2}$ are solutions of the homogeneous equation! (Basically this means that your generic guess for $y_p$ has to be multiplied by a power of $x$.) Finally, add the general solution of the homogeneous equation (which has two arbitrary constants) to the particular solution you found using undetermined coefficients.
P.S. Since the two lowest degree terms of the input function, $2e^{-x/2}$ and $6xe^{-x/2}$, are also solutions of the homogeneous equation, you need to throw a factor of $x^2$ into your form for $y_p$, so it should be
$$y_p(x)=(Ax^4+Bx^3+Cx^2)e^{-x/2}.$$
