# Method of Variation of Parameters

• ElijahRockers
In summary, the conversation discusses solving a differential equation using the characteristic equation and calculating the Wronskian. The solution involves finding the homogenous solution and using the formula for u1. There is some confusion about integrating by parts and a mistake in the calculation of the roots. The correct solution involves finding the Wronskian to be e^(-2x) and correcting the roots to be +1.
ElijahRockers
Gold Member

## Homework Statement

$y''-2y'+y = \frac{e^x}{1+x^2}$

## Homework Equations

$u_1 = -\int \frac{y_{2}g(x)}{W}dx$
$u_2 = \int \frac{y_{1}g(x)}{W}dx$
$g(x) = \frac{e^x}{1+x^2}$
W is the wronskian of y1 and y2.

## The Attempt at a Solution

The characteristic equation for the homogenous solution yields a repeated root of -1, so

$y_{h} = C_{1}e^{-x} + C_{2}xe^{-x}$

When I calculated the Wronskian it simplified to

$W = e^{-2x}$

Plugging in the formula for u1 and simplifying I get

$u_{1} = -\int \frac{xe^{2x}}{1+x^2}dx$

I'm not quite sure how to go about solving this. My best shot at integration by parts didn't really seem to help. I ended up having to integrate ln(x^2 + 1)e^2x which doesn't seem any easier.

I got W=(1-2x)e-2x.

Hmmm, alright.. I can't seem to find my mistake

if W is
|y1 y2|
|y'1 y'2|

I got
y1 = e^-x
y2 = xe^-x
y'1 = -e^-x
y'2 = -xe^-x + e^-x = e^-x(1-x) (after factoring off the e^-x)

and y1y'2 - y'1y2 = e^(-2x)(1-x) + xe^(-2x) = e^(-2x)

Sorry, I made a sign mistake.

Oh, your roots are wrong. They should be +1.

O geez... always something little.

Thanks!

## What is the Method of Variation of Parameters?

The Method of Variation of Parameters is a mathematical technique used to solve ordinary differential equations by finding a particular solution based on the complementary function. It involves using a set of functions as a basis for the particular solution and then solving for the coefficients of those functions.

## When is the Method of Variation of Parameters used?

The Method of Variation of Parameters is typically used when the complementary function of a differential equation is known, but a particular solution cannot be found using traditional methods such as the Method of Undetermined Coefficients or the Method of Reduction of Order.

## What are the steps involved in using the Method of Variation of Parameters?

The steps involved in using the Method of Variation of Parameters are as follows:

• Step 1: Find the complementary function of the differential equation
• Step 2: Choose a set of functions as a basis for the particular solution
• Step 3: Substitute the basis functions into the original differential equation
• Step 4: Solve for the coefficients of the basis functions
• Step 5: Combine the complementary function and the particular solution to obtain the general solution

## What are the advantages of using the Method of Variation of Parameters?

The Method of Variation of Parameters allows for the solution of differential equations that cannot be solved using traditional methods. It also provides a more systematic approach to finding particular solutions compared to other techniques.

## Are there any limitations to the Method of Variation of Parameters?

One limitation of the Method of Variation of Parameters is that it can only be applied to linear differential equations with constant coefficients. It also requires some knowledge of the complementary function, which may not always be known or easy to find.

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