91Ó°ÊÓ

When we face a differential equation like \(y'' + y' - 6y = 0\), one of the first steps is finding its characteristic equation. The characteristic equation is essential as it transforms a differential equation into a simpler algebraic equation, allowing us to find solutions more easily.
To get there, we assume a solution of the form \(y = e^{rt}\). This assumption allows the derivatives to be expressed in terms of \(r\): \(y' = re^{rt}\) and \(y'' = r^2e^{rt}\). Substituting these into the differential equation forms an equation in terms of \(e^{rt}\).
The equation reads: \[r^2e^{rt} + re^{rt} - 6e^{rt} = 0\]. Since \(e^{rt}\) is never zero, we can simplify it to \(r^2 + r - 6 = 0\). This is our characteristic equation. Solving this equation helps determine the values of \(r\) that satisfy the original differential equation.

The quadratic formula is a handy tool used to solve the characteristic equation \(r^2 + r - 6 = 0\), which is a quadratic equation. A quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(r\) is the variable
.

• For our equation, \(a = 1\), \(b = 1\), and \(c = -6\).

Please note that the formula is \[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Here is how we solve the equation:
1. Compute \(b^2 - 4ac\): \(1^2 - 4 \cdot 1 \cdot (-6) = 1 + 24 = 25\).
2. Take the square root of 25, which is 5.
3. Substitute the values into the formula:
\[r = \frac{-1 \pm 5}{2}\].
This results in two potential solutions, \(r_1 = 2\) and \(r_2 = -3\). These solutions are crucial to building the general solution for the differential equation.

Once we have the roots from the characteristic equation (\(r_1 = 2\) and \(r_2 = -3\)), we can easily find the general solution of the differential equation. This involves creating a linear combination of solutions corresponding to each root.
The general solution will therefore be:

• \(y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\)
• Substituting the roots, we get: \(y(t) = C_1 e^{2t} + C_2 e^{-3t}\)

Here, \(C_1\) and \(C_2\) are arbitrary constants, which can be determined if we have initial conditions. This form is typically called the homogeneous solution and forms the basis for solving many differential equations by applying relevant initial or boundary conditions. Each of the terms in the general solution addresses parts of the behavior of the system described by the differential equation.