2nd ODE constant coeff, quick question

[rock.freak667]

Homework Statement

Solve:
$$\frac{d^2y}{dx^2}+\frac{dy}{dx}=3x^2+2x+1$$

The Attempt at a Solution

Well the C.F. is $y=C_1e^{-x}$
the P.I. is of the form $y_{PI}=Ax^3+Bx^2+Cx+D$

I can find the values of A,B and C bu differentiating it and substituting it into the equation. But How would I find D since there is no 'y' in the ODE given and differentiating $y_{PI}$ makes the constant disappear.
(Note: I can find the answer by integrating it w.r.t x and then using the integrating factor but I would like to know how to find it by adding the PI and CF together)

 

[John Creighto]

Homework Statement

Solve:
$$\frac{d^2y}{dx^2}+\frac{dy}{dx}=3x^2+2x+1$$

The Attempt at a Solution

Well the C.F. is $y=C_1e^{-x}$
the P.I. is of the form $y_{PI}=Ax^3+Bx^2+Cx+D$

I can find the values of A,B and C bu differentiating it and substituting it into the equation. But How would I find D since there is no 'y' in the ODE given and differentiating $y_{PI}$ makes the constant disappear.



(Note: I can find the answer by integrating it w.r.t x and then using the integrating factor but I would like to know how to find it by adding the PI and CF together)

The D is redundant since one of the solutions to the natural equation is a constant function.

Therefor the constant function is determined by the initial conditions rather then the forcing function.

 

[rock.freak667]

Ah...thanks then,I thought I was really doing something wrong.