# Variation of Parameters/Wronskian

• smashyash
In summary, the conversation discusses factoring using characteristics to solve a problem involving a second-order linear differential equation. The process involves finding the complementary solution using the characteristic equation, and then using the Wronskian to find the particular solutions. However, the use of the Wronskian may not be necessary in some cases, as shown by the answer given in the back of the book.
smashyash
So I'm doing some practice problems to prepare for a test on Friday and I'm just curious about this problem::

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

in factoring using characteristics:

(r+2)(r+1) = 0
r = -2,-1

so Yc = C1*e^(-2x) + C2*e^(x)
y1= e^(-2x)
y2= e^(-x)

(skipping some algebra)..I used these to find u1' and u2':

u1' = -4e^(3x)
u2' = 4e^(2x)

Now, I thought that this was the part where you use y1, y1', y2, y2' do find the wronskian and divide u1' and u2' by w(y1,y2) but apparently no because the answer given in the back of the book tells me that the wronskian isn't even used... why is that? I thought the wronskian was included in the equation::

Yp = -y1(x) * INT( (y2(x)*f(x) ) / W(x) ) dx + y2(x) INT( (y1(x)*f(x) ) / W(x) ) dx

Maybe I'm not understanding this equation as well as I thought I did.. Any comments?? Thanks!

You use the Wronskian to find your u1' and u2':

u1' = - y2*f(x)/W
u2' = y1*f(x)/W

Then just integrate them to get the particular solutions.

## 1. What is the Variation of Parameters method?

The Variation of Parameters method is a technique used to find a particular solution to a non-homogeneous linear differential equation. It involves finding a set of functions that satisfy the non-homogeneous equation and using them to construct a particular solution.

## 2. How is the Wronskian used in the Variation of Parameters method?

The Wronskian is a determinant that is used to determine if a set of functions is linearly independent. In the Variation of Parameters method, the Wronskian is used to ensure that the functions chosen for the particular solution are linearly independent, which is necessary for the method to work.

## 3. Can the Variation of Parameters method be used for all non-homogeneous linear differential equations?

No, the Variation of Parameters method can only be used for non-homogeneous linear differential equations with constant coefficients. It cannot be applied to equations with variable coefficients or non-linear equations.

## 4. What are the steps involved in using the Variation of Parameters method?

The first step is to solve the associated homogeneous equation to find a set of linearly independent solutions. Then, the Wronskian is used to determine a particular solution by finding a set of functions that satisfy the non-homogeneous equation. Finally, the particular solution is combined with the solutions from the homogeneous equation to form the general solution.

## 5. Are there any limitations or drawbacks to using the Variation of Parameters method?

One limitation of the Variation of Parameters method is that it can be time-consuming and complex, especially for higher-order differential equations. Additionally, it may not always be possible to find a set of linearly independent solutions, in which case the method cannot be applied.

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