91Ó°ÊÓ

Understanding the characteristic equation is essential when solving homogeneous second-order linear differential equations like the one given: 6\( y'' - 5 y' - 6 y = 0 \). Here, the approach involves looking for solutions of the form \( y = e^{rt} \). The reason this form is chosen is due to its simplicity and because derivatives of exponential functions are also exponentials. Substituting \( y = e^{rt} \) into the differential equation, you replace \( y, y' \) and \( y'' \) with \( e^{rt} \), \( re^{rt} \), and \( r^2e^{rt} \) respectively. This substitution leads to the characteristic equation:

• \( 6r^2 - 5r - 6 = 0 \)

The roots of this polynomial equation provide the values of \( r \) that will form part of the general solution of the differential equation. Solving the characteristic equation is crucial, as the nature of its roots (whether real or complex) determines the form of the solution.

To solve the characteristic equation, \( 6r^2 - 5r - 6 = 0 \), the quadratic formula is used. The quadratic formula is a tried-and-true method pronounced as follows:

• \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The particular values for this exercise are \( a = 6 \), \( b = -5 \), and \( c = -6 \). The discriminant, \( b^2 - 4ac \), is a pivotal part of this formula:

• \( (-5)^2 - 4(6)(-6) = 25 + 144 = 169 \)

A positive discriminant indicates two distinct real roots. Calculating further, we find:

• \( r = \frac{5 \pm 13}{12} \) yielding roots \( r_1 = \frac{3}{2} \) and \( r_2 = -\frac{2}{3} \).

These roots are essential for forming the general solution of the differential equation.

Once you have the roots of the characteristic equation, you can write the general solution to a second-order differential equation. In this scenario, since the characteristic equation yields two distinct real roots, \( r_1 = \frac{3}{2} \) and \( r_2 = -\frac{2}{3} \), the general solution adopts the following form:

• \( y(t) = C_1 e^{\frac{3}{2}t} + C_2 e^{-\frac{2}{3}t} \)

Here, \( C_1 \) and \( C_2 \) are arbitrary constants, which are determined by initial or boundary conditions if provided. This formula represents a superposition of solutions, where each exponential component corresponds to one of the roots. The concept of the general solution highlights the importance of the characteristic equation's roots, as they directly affect the behavior of the solution curve over time. In practical terms, this reflects different dynamic behaviors of systems modeled by such differential equations.