Method of undetermined coefficients -- Help please

In summary, the method of undetermined coefficients can be used to determine the general solution of a second-order, linear, non-homogenous equation. In this specific case, the equation is y'' - 4y' + 4y = 2e^(2x+3). By solving for r in the equation r^2 - 4r + 4 = 0, it is found that r = 2. Therefore, the solution is in the form of y = C1e^(2x) + C2xe^(2x). To find the particular solution, Yp = Ax^(2)e^(2x+3) and after taking the derivatives, plugging them into the equation and simplifying
  • #1
phantom lancer
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Homework Statement


Using the method of undetermined coefficients, determine the general solution of the following second-order, linear, non-homogenous equations.

y'' - 4y' + 4y = 2e^(2x+3)

Homework Equations


I'm not sure what to do from here...
Also, I'm new here. How do I use the superscript for exponents?

The Attempt at a Solution


r^2 - 4r + 4 = 0
r = 2, so y = C1e^(2x) + C2xe^(2x)

I assume Yp = Ax^(2)e^(2x+3)
so, Yp' = 2Ax^(2)e^(2x+3)
Yp'' = 4Ax^(2)e^(2x+3)

plugging them in the equation: 4Ax^(2)e^(2x+3) - 4(2Ax^(2)e^(2x+3) + 4(Ax^(2)e^(2x+3) = Ax^(2)e^(2x+3)

I get 0 = Ax^(2)e^(2x+3)

From here, I don't know what to do. Please Help.
 
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  • #3
There are the little x2 above the typing box. When you have two functions multiplied, you need to use the Product rule to take the derivative.
 
  • #4
In your differentiation of the particular solution you are forgetting the term that comes from differentiating the ##x^2##.
 
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Related to Method of undetermined coefficients -- Help please

1. What is the method of undetermined coefficients?

The method of undetermined coefficients is a technique used to solve non-homogeneous linear differential equations with constant coefficients. It involves finding a particular solution by assuming the form of the solution and then solving for the coefficients.

2. When is the method of undetermined coefficients used?

The method of undetermined coefficients is typically used when the non-homogeneous term in the differential equation is a polynomial, exponential, sine, or cosine function. It is not suitable for more complex functions such as logarithmic or trigonometric functions.

3. How do you apply the method of undetermined coefficients?

To apply the method of undetermined coefficients, you first need to identify the form of the particular solution based on the non-homogeneous term. Then, substitute the solution into the differential equation and solve for the undetermined coefficients. Finally, combine the particular solution with the general solution of the corresponding homogeneous equation to obtain the complete solution.

4. What is the difference between the method of undetermined coefficients and variation of parameters?

The method of undetermined coefficients is a specific technique used to solve non-homogeneous linear differential equations with constant coefficients. Variation of parameters, on the other hand, is a more general method that can be used to solve non-homogeneous differential equations with variable coefficients.

5. Are there any limitations to the method of undetermined coefficients?

Yes, the method of undetermined coefficients has some limitations. It can only be used for certain types of non-homogeneous terms and cannot handle more complex functions. Additionally, it may not work if the non-homogeneous term is similar to one of the terms in the complementary function.

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