# 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.

• The solution of the homogeneous equation must be very inspiring to you. Nov 14, 2014 at 6:12
• From the complementary function, how do I go to the particular integral? Nov 14, 2014 at 6:14
• What is the solution of the homogeneous equation ? Nov 14, 2014 at 6:15
• What did you get for the complementary function? It's hard to get a particular solution without knowing that.
– bof
Nov 14, 2014 at 6:16
• I don't know a way to find a particular integral without knowing the complementary function. I could work out the complementary function, but please don't make me repeat the work you've already done.
– bof
Nov 14, 2014 at 6:18

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: $y = y_h+y_p$

• I need the particular integral.. Nov 14, 2014 at 6:11
• @user2774555 If all you need is a particular solution, you should have said so in your question. And you should show the work you have already done, so we know what kind of help you need. What did you get for the homogeneous solution? And what did you try for a particular solution?
– bof
Nov 14, 2014 at 6:14
• @OC-Sansoo Thanks! Nov 14, 2014 at 6:17
• @bof I had tried the problem, but I was not sure about the method of undetermined coefficients, which I read about just now. So now I can understand that. Thanks.. Nov 14, 2014 at 6:18
• Are you sure of your form for $y_p$? I'd try multiplying it by $x^2$.
– bof
Nov 14, 2014 at 6:19

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 $$y_p(x)=(Ax^4+Bx^3+Cx^2)e^{-x/2}.$$