91Ó°ÊÓ

When solving second-order linear homogeneous differential equations, the characteristic equation is a key concept. It acts as a bridge between the differential equation and algebra. This equation is formed by replacing the derivatives in the differential equation with powers of a variable, usually denoted as \( r \). For instance, in the equation \( y'' - y' - 2y = 0 \), we replace \( y'' \) with \( r^2 \), \( y' \) with \( r \), and \( y \) with 1, resulting in the characteristic equation \( r^2 - r - 2 = 0 \).
This transformation is possible because each derivative corresponds directly to the power of \( r \) that acts as a placeholder for our solutions. Solving the characteristic equation will give us the roots, which are essential for finding the general solution to the differential equation.

The quadratic formula is a simple and effective method for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is expressed as \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). By substituting the correct values of \( a \), \( b \), and \( c \) from the characteristic equation into this formula, we can find its roots.
In our example, after identifying \( a = 1 \), \( b = -1 \), and \( c = -2 \), we substitute these into the quadratic formula, leading to \( r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)} \). Simplifying this gives us \( r_1 = 2 \) and \( r_2 = -1 \). Using the quadratic formula ensures a reliable way to find these solutions efficiently, regardless of the specific coefficients.

The general solution of a differential equation provides a family of solutions that satisfy the equation for any constants included within it. For second-order linear homogeneous differential equations with real and distinct roots, the general solution takes the form: \( y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} \).
This expression combines the exponential functions determined by the roots of the characteristic equation.
Using the roots \( r_1 = 2 \) and \( r_2 = -1 \), our general solution becomes \( y(x) = C_1 e^{2x} + C_2 e^{-x} \). Here, \( C_1 \) and \( C_2 \) are arbitrary constants, which will be adjusted based on initial conditions or other constraints of the specific problem.

Real and distinct roots occur when the discriminant \( b^2 - 4ac \) in the quadratic formula is positive. For the characteristic equation \( r^2 - r - 2 = 0 \), the discriminant is \( 9 \), which is positive, yielding roots \( r_1 = 2 \) and \( r_2 = -1 \).
These roots indicate that our general solution will have two different exponential terms. Each root corresponds to an exponential function component \( e^{r_1 x} \) and \( e^{r_2 x} \).
This property ensures that the general solution is a combination of independent solutions, giving it more flexibility to satisfy various initial conditions.