# Method of undetermined coefficients question

• illidari
In summary, The particular solution for g(x) = x^2 in the differential equation y'' + y' = g(x) is yp1 = x(Ax^2 + Bx + C), where A, B, and C are constants and x is multiplied to avoid duplicating terms in the fundamental set of solutions.
illidari

## Homework Statement

y''+y'=g(x)
fundamental set of solns. of the homog. DE is:
y1= 1 , y2= e^-x

If g(x) = x^2 , yp1 = ____

## The Attempt at a Solution

I actually am looking at a key to an old test my teacher gave as a review for a test. The key has the asnswer as x(Ax^2+Bx+C)

Is there any reason why the x was added? Shouldn't it only be (Ax^2+Bx+C). Much thanks if someone can clarify this for me.

I'm pretty sure its because when you're finding your particular solution, you can't start with terms that duplicate terms in the fundamental set. Since you have a constant as a solution in the fundamental set, you need to multiply the particular solution by x (because of C).

## 1. What is the Method of Undetermined Coefficients?

The Method of Undetermined Coefficients is a technique used in solving differential equations with constant coefficients. It involves assuming a particular form for the solution and determining the coefficients by plugging it into the original equation.

## 2. When is the Method of Undetermined Coefficients used?

The Method of Undetermined Coefficients is typically used when the non-homogeneous term in a differential equation is a polynomial, exponential, sine, or cosine function. It is not applicable for more complicated functions such as logarithmic or trigonometric functions.

## 3. How does the Method of Undetermined Coefficients work?

The method involves two steps: first, assuming a particular form for the solution based on the non-homogeneous term, and second, substituting the assumed solution into the original equation and solving for the coefficients.

## 4. What happens if the assumed solution is also a solution of the homogeneous equation?

If the assumed solution is also a solution of the homogeneous equation, a modification needs to be made to the assumed solution before substituting it into the original equation. This is done by multiplying the assumed solution by the variable x, until it is no longer a solution of the homogeneous equation.

## 5. Are there any limitations to using the Method of Undetermined Coefficients?

Yes, the method can only be used for non-homogeneous equations with constant coefficients and specific types of non-homogeneous terms. It also may not work for all cases, in which case other methods such as Variation of Parameters may need to be used.

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