What are the steps for finding u1 and u2 in the variation of parameters method?

In summary, the conversation discusses finding the general solution for a mathematical problem involving a characteristic equation and a system of equations with trigonometric functions. The conversation also includes a question about integration and a reference request for solving systems of equations involving trigonometric equations.
  • #1
Cafka
6
0
Hello,
I'm trying to understand this concept. Jere's the problem I'm doing.
I have to find the general solution for:
y'' + 36y = -4xsin(6x)

So you then solve for your characteristic equation and get lamda = +/- 6
so y1 = e^-6x and y2 = e^6x
You get your matrix for w, w1, and w2.
w = 12
w1 = 4xe^6xsin(6x)
w2 = -4xe^-6xsin(6x)

I have the problem at getting u1 and u2.
u1 = 1/3 the integral of xe^6xsin(6x) dx
u2 = -1/3 the integral of xe^-6xsin(6x) dx

How do you do that integration?

Thanks for your help,
Tom
 
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  • #2
Your characteristic equation should be of the form y=Asin6x+Bcos6x since the the squareroot of -36 is 6i and -6i =)
 
  • #3
Yeah,and Lagrange's method kicks ass.So apply it.

Daniel.
 
  • #4
I was wondering if anyone knew of any good references on solving systems of equations involving trigonometric equations. Any information would be appreciated.
 
  • #5
Found the answer - I had forgotten Cramer's rule.
 

FAQ: What are the steps for finding u1 and u2 in the variation of parameters method?

1. What is the concept of "variation of parameters"?

"Variation of parameters" is a method used in solving differential equations. It involves finding a particular solution by assuming it has the form of a linear combination of known functions, and then solving for the coefficients of those functions.

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

This method is typically used when the differential equation is inhomogeneous, meaning it has a non-zero forcing term. It is also used when the coefficients of the differential equation are not constant.

3. How does the "variation of parameters" method differ from other methods of solving differential equations?

The "variation of parameters" method differs from other methods, such as the method of undetermined coefficients, because it allows for the solution to be expressed as a linear combination of known functions, rather than just a single function. This makes it more versatile and applicable to a wider range of differential equations.

4. What are the steps involved in using the "variation of parameters" method?

The steps involved in using the "variation of parameters" method are:
1. Find the general solution of the associated homogeneous equation.
2. Assume a particular solution in the form of a linear combination of known functions.
3. Substitute the assumed solution into the original differential equation and solve for the coefficients.
4. Add the particular solution to the general solution of the homogeneous equation to get the general solution of the nonhomogeneous equation.

5. Can the "variation of parameters" method be used for higher order differential equations?

Yes, the "variation of parameters" method can be used for higher order differential equations. However, the process becomes more complex as the order of the equation increases, as more unknown coefficients need to be solved for. It is also important to note that this method is more commonly used for first and second order differential equations.

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