Simple question about Variaton of Parameters in 1 variable

In summary, when writing the Wronskian for a generic second order ODE, W1 is obtained by replacing the first column of the determinant with zeroes until reaching the end, where it is replaced with g(x). This method, known as Cramer's rule, can be used to solve a system of equations by forming fractions with the determinant of coefficients in the denominator and the right hand side of the system in the numerator. This method can be applied to any initial value problem as long as a solution exists.
  • #1
Remixex
57
4
Given a generic second order ODE, normalized that equals to g(x)
I saw in a video that, when writing the Wronskian, W1 and W2, for writing W1 you had to "delete" the first column if the Determinant and replace it with zeroes until you reach the end, where you replace it with g(x)
Is this always true no matter the initial value problem? (as long as the solution exists of course)
 
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  • #2
I think you are misunderstanding the purpose. That sounds like it is not directly a matter of "variation of parameters" but specifically a method of solving a system of equations called "Cramer's rule". To find the "i"th unknown in a system of n equations form a fraction with the determinant of coefficients in the denominator and the numerator the same except that the i column is replaced by the right hand side of the system.
For example, if ax+ by = e and cx+ dy= f then
[tex]x= \frac{\left|\begin{array}{cc} e & b \\ f & d \end{array}\right|}{\left|\begin{array}{cc}a & b \\ c & d \end{array}\right|}[/tex]
and
[tex]y= \frac{\left|\begin{array}{cc}a & e \\ c & f \end{array}\right|}{\left|\begin{array}{cc}a & b \\ c & d \end{array}\right|}[/tex]
 

Related to Simple question about Variaton of Parameters in 1 variable

1. What is the concept of Variation of Parameters in 1 variable?

Variation of Parameters in 1 variable is a method used in differential equations to find a particular solution. It involves using a trial solution that contains arbitrary constants and substituting it into the original equation to determine the values of those constants.

2. When is Variation of Parameters used?

Variation of Parameters is typically used when the coefficients in a differential equation are not constant. It is also used when the method of undetermined coefficients is not applicable, such as when the non-homogeneous term is a polynomial of the same degree as the complementary function.

3. How does Variation of Parameters differ from the method of undetermined coefficients?

The key difference between the two methods is that Variation of Parameters involves using a trial solution with arbitrary constants, while the method of undetermined coefficients uses a specific form of the non-homogeneous term to find a particular solution. Variation of Parameters is more general and can be used in a wider range of cases.

4. Are there any limitations to using Variation of Parameters?

Yes, there are some limitations to using Variation of Parameters. It can only be used for linear differential equations with constant coefficients and non-homogeneous terms that are functions of a single variable. It also requires the complementary function to be known beforehand.

5. Can you provide an example of using Variation of Parameters to solve a differential equation?

Sure, let's say we have the differential equation y'' + 2y' + y = 2x + 1. We can use Variation of Parameters by first finding the complementary function y_c = c_1e^{-x} + c_2xe^{-x}. Then, we can use a trial solution of y_p = u_1(x)e^{-x} + u_2(x)xe^{-x}, where u_1(x) and u_2(x) are arbitrary functions. Plugging this into the original equation and solving for u_1(x) and u_2(x) will give us the particular solution y_p = -2x + \frac{1}{2}x^2 + \frac{5}{4}e^{-x}.

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