# Variation of parameters to obtain PS of 2nd Order non-hom equation

• robot1000
In summary, the conversation discusses solving a question involving a differential equation and obtaining a general solution. The Wronskian is calculated and the variation of parameters formula is used, resulting in a solution that may be incorrect. The person suggests using numerical solutions to validate the analytic solution.
robot1000
The question I'm trying to solve is:

y" - 6y' + 9y = $\frac{exp(3x)}{(1+x)}$

I formulated the Gen solution which are:

y1(x) = exp(3x) and y2(x) = xexp(3x)

I've then calculated the wronskian to get: exp(6x)

I then went onto to use the variation of parameters formula, which is where I got a bit stuck

I ended up with

-exp(3x)*(x - ln(x+1) + xexp(3x)*ln(1+x)

The problem is, it just doesn't look right.

I would appreciate some guidance with this problem

Except for a parentheses that you missed to close the (x - ln(x+1)), it looks right to me. Why would you say that it doesn't look right, and what guidance do you expect to get?

robot1000 said:
I would appreciate some guidance with this problem

Here's what you do. You solve it numerically first and then plot the analytic solution you get over the numeric solution. If they agree, right on top of one another, then there is very good odds your analytic solution is correct. If you're going to work with DEs, this is a very useful practice in my opinion. So learn how to set all this up in Mathematica or another CAS and you will never say again, "that don't look right."

## 1. What is the variation of parameters method?

The variation of parameters method is a technique used to solve second-order non-homogeneous differential equations. It involves finding a particular solution by assuming it to be a linear combination of two linearly independent solutions to the corresponding homogeneous equation.

## 2. When is the variation of parameters method used?

The variation of parameters method is used when solving second-order non-homogeneous differential equations that cannot be solved using other techniques such as the method of undetermined coefficients or the annihilator method.

## 3. How does the variation of parameters method work?

The variation of parameters method involves finding the general solution to the corresponding homogeneous equation, then using it to find a particular solution by replacing the constants with functions of the independent variable. The final solution is a linear combination of the general solution and the particular solution.

## 4. What are the advantages of using the variation of parameters method?

One advantage of using the variation of parameters method is that it can be applied to a wide range of second-order non-homogeneous differential equations. Additionally, it does not require any prior knowledge of the form of the particular solution, unlike other methods.

## 5. Are there any limitations to the variation of parameters method?

One limitation of the variation of parameters method is that it can become quite complex and time-consuming for higher-order differential equations. It also requires a good understanding of linear algebra and the theory of differential equations.

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