91Ó°ÊÓ

In the context of solving differential equations, the characteristic equation is a critical concept. It's directly linked to a linear differential equation with constant coefficients and is essential for finding the solution. When we have a second-order differential equation like the one given:
\(2 y^{\prime \prime}+13 y^{\prime}-7 y=0\), the characteristic equation provides a polynomial form that helps in solving the differential equation.
To derive this equation, we replace the derivatives of \(y\), such as \(y'\) and \(y''\), with terms involving \(r\), where \(r\) represents a root we wish to find. This conversion changes the differential equation into

• \(2r^2 + 13r - 7 = 0\)

Solving this characteristic polynomial allows us to find \(r\), the values of which dictate the behavior of the system described by the differential equation.

The quadratic formula is a universal method used to solve for roots when we have any quadratic equation in the form \(ax^2 + bx + c = 0\). Often, quadratic equations aren't factorable using simple methods, so the quadratic formula becomes indispensable.
For the characteristic equation \(2r^2 + 13r - 7 = 0\), we apply the quadratic formula:

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

Substituting \(a = 2\), \(b = 13\), and \(c = -7\), we get

• \(r = \frac{-13 \pm \sqrt{13^2 - 4 \times 2 \times (-7)}}{2 \times 2}\)

The discriminant here, \(b^2 - 4ac\), determines the nature of the roots:

• If it's positive, there are two real roots.
• If zero, there's one real root.
• If negative, the roots are complex.

In this case, calculating the discriminant will show if the equation has real or complex solutions.

Once the roots \(r_1\) and \(r_2\) of the characteristic equation are found, we proceed to construct the general solution of the differential equation. This is imperative because it provides the form that describes all possible solutions to the differential system.
For second-order differential equations with constant coefficients, the general solution has the form:

• \(y = c_1 e^{r_1 x} + c_2 e^{r_2 x}\)

Here, \(c_1\) and \(c_2\) are arbitrary constants determined by initial conditions or boundary values of the problem you're solving. The terms \(e^{r_1 x}\) and \(e^{r_2 x}\) arise because the solutions to the characteristic equation determine exponential functions that align with our differential equation form.
This structure allows for expressing the complexity of the system dynamics, often describing physical phenomena or engineering systems as a combination of exponential behaviours.