12. Persamaan Diferensial Biasa Orde Tinggi

Introduction

In this tutorial, we will explore the solution of higher-order linear ordinary differential equations (ODEs) by finding their characteristic equations and factoring them. This approach is crucial for solving linear differential equations efficiently. We'll also look into methods for factoring polynomials of degree higher than two using Horner's method or synthetic division.

Step 1: Understanding Higher-Order Linear ODEs

• A higher-order linear ordinary differential equation is one that involves derivatives of an unknown function up to a certain order.
• The general form of an n-th order linear ODE can be expressed as: [ a_n y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_1 y' + a_0 y = f(x) ] where (y) is the unknown function, (a_i) are coefficients, and (f(x)) is a function of (x).

Step 2: Find the Characteristic Equation

• To solve the ODE, first, we need to derive the characteristic equation:
  • Replace (y) with (r^n) (where (r) is a constant) and derivatives with powers of (r).
  • The characteristic equation is then: [ a_n r^n + a_{n-1} r^{n-1} + ... + a_1 r + a_0 = 0 ]

Step 3: Factor the Characteristic Equation

• After obtaining the characteristic equation, the next step is to factor it. This can be done using:
  • Horner's Method: A systematic way to evaluate and factor polynomials.
  • Synthetic Division: A simplified form of polynomial long division.

Using Horner's Method

1. Write the coefficients of the polynomial in order.
2. Choose a value for (r) (a possible root) and perform the synthetic division.
3. Repeat until all roots are found or a degree of 1 is reached.

Using Synthetic Division

1. Identify a potential root (often guessed or found using the Rational Root Theorem).
2. Divide the polynomial by ( (r - \text{root}) ).
3. Continue dividing the resulting polynomial until it's fully factored.

Step 4: Solve for Roots

• Once factored, set each factor equal to zero to solve for (r): [ (r - r_1)(r - r_2)...(r - r_n) = 0 ]
• This will give you the roots needed for constructing the general solution of the ODE.

Step 5: Construct the General Solution

• Depending on the nature of the roots (real and distinct, real and repeated, or complex), construct the general solution:
  • Distinct Real Roots: Combine terms of the form (C e^{r_ix}).
  • Repeated Roots: Combine terms of the form (C_1 e^{r_ix} + C_2 x e^{r_ix}).
  • Complex Roots: Use Euler's formula to express in terms of sine and cosine.

Conclusion

In summary, solving higher-order linear ordinary differential equations involves finding the characteristic equation, factoring it, solving for roots, and constructing the general solution based on those roots. By mastering these steps, you can tackle a wide range of linear ODEs effectively. Next, consider practicing with various equations to strengthen your understanding and application of these methods.