Berikut adalah artikel tentang metode koefisien tak tentu untuk solusi PDB orde dua:
The Undetermined Coefficients Method for Solving Second-Order Linear Non-Homogeneous Differential Equations
The method of undetermined coefficients is a powerful technique used to find particular solutions to second-order linear non-homogeneous differential equations. This method is particularly effective when the non-homogeneous term (the forcing function) is a relatively simple function, such as a polynomial, exponential, sine, cosine, or a combination thereof. Let's delve into the details.
Understanding the Structure of the Equation
We're dealing with differential equations of the form:
ay'' + by' + cy = g(x)
where:
- 'a', 'b', and 'c' are constants.
- 'y'' and 'y'' represent the second and first derivatives of 'y' with respect to 'x', respectively.
- 'g(x)' is the non-homogeneous term (forcing function).
The general solution to this equation is the sum of the complementary solution (y<sub>c</sub>) and the particular solution (y<sub>p</sub>):
y = y<sub>c</sub> + y<sub>p</sub>
We already know how to find y<sub>c</sub> using techniques like finding the roots of the characteristic equation. This article focuses on efficiently determining y<sub>p</sub> using the method of undetermined coefficients.
The Undetermined Coefficients Method: A Step-by-Step Guide
This method relies on making an educated guess about the form of the particular solution based on the form of g(x). Hereβs a breakdown:
1. Identify the Form of g(x)
This is the crucial first step. The table below provides a guide:
| g(x) | Form of y<sub>p</sub> |
|---|---|
| k (constant) | A |
| k x<sup>n</sup> | A<sub>0</sub> + A<sub>1</sub>x + A<sub>2</sub>x<sup>2</sup> + ... + A<sub>n</sub>x<sup>n</sup> |
| k e<sup>rx</sup> | Ae<sup>rx</sup> |
| k cos(Οx) or k sin(Οx) | A cos(Οx) + B sin(Οx) |
| k e<sup>rx</sup>cos(Οx) or k e<sup>rx</sup>sin(Οx) | e<sup>rx</sup>(A cos(Οx) + B sin(Οx)) |
Note: 'A', 'B', 'A<sub>0</sub>', 'A<sub>1</sub>', etc. are undetermined coefficients that we'll solve for.
2. Adjust for Duplication
If any term in your assumed y<sub>p</sub> is already a solution to the complementary equation (y<sub>c</sub>), you must modify your guess. Multiply the term by x (or x<sup>2</sup> if it's still a solution).
3. Substitute and Solve
Substitute your assumed y<sub>p</sub> into the original differential equation. This will give you an equation involving 'x'. Equate the coefficients of corresponding terms to solve for the undetermined coefficients (A, B, etc.).
4. Write the General Solution
Finally, combine your calculated y<sub>p</sub> with the complementary solution y<sub>c</sub> to get the complete general solution: y = y<sub>c</sub> + y<sub>p</sub>.
Example: Solving a Second-Order Differential Equation
Let's solve the equation: y'' - 3y' + 2y = 4x
-
Find y<sub>c</sub>: The characteristic equation is r<sup>2</sup> - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. Thus, r<sub>1</sub> = 1 and r<sub>2</sub> = 2. Therefore, y<sub>c</sub> = C<sub>1</sub>e<sup>x</sup> + C<sub>2</sub>e<sup>2x</sup>.
-
Find y<sub>p</sub>: Since g(x) = 4x, we assume y<sub>p</sub> = Ax + B.
-
Substitute and Solve: Substituting into the original equation and simplifying gives: -3A + 2(Ax + B) = 4x. Equating coefficients yields: 2A = 4 (coefficient of x) and -3A + 2B = 0 (constant term). Solving, we find A = 2 and B = 3. Therefore, y<sub>p</sub> = 2x + 3.
-
General Solution: The general solution is y = y<sub>c</sub> + y<sub>p</sub> = C<sub>1</sub>e<sup>x</sup> + C<sub>2</sub>e<sup>2x</sup> + 2x + 3.
Conclusion
The method of undetermined coefficients is a valuable tool for solving a specific class of second-order linear non-homogeneous differential equations. While it requires careful attention to detail and a systematic approach, mastering this technique is crucial for students and professionals working with differential equations. Remember to always check for duplication between y<sub>c</sub> and your initial guess for y<sub>p</sub>. Practice and understanding the underlying principles will lead to greater proficiency in applying this powerful technique.
