91Ó°ÊÓ

When we have a second-order linear homogeneous differential equation like the given problem, we start by finding the characteristic equation. This is a crucial step that transforms the differential equation into an algebraic one, making it easier to handle. For the given differential equation: \(y'' - 8y' + 15y = 0\), we write its characteristic equation by replacing \(y''\) with \(r^2\), \(y'\) with \(r\), and \(y\) with 1. This turns it into: \(r^2 - 8r + 15 = 0\). Writing the characteristic equation in this manner simplifies the solving process, setting us up to find the roots.

The characteristic equation we obtain is a quadratic equation. A quadratic equation is a polynomial equation of degree 2, and it generally looks like: \(ax^2 + bx + c = 0\). In our case, it is: \(r^2 - 8r + 15 = 0\). Solving quadratic equations can be done using various methods such as factoring, completing the square, or using the quadratic formula. Here, we use factoring, which involves rewriting the quadratic equation as a product of its linear factors. For our problem, we factorize \(r^2 - 8r + 15\) as: \((r - 3)(r - 5) = 0\). Factoring is a powerful technique because it immediately gives us the roots when each factor is set to zero: \(r - 3 = 0\) and \(r - 5 = 0\), leading to \(r = 3\) and \(r = 5\), respectively.

With the roots from the factored characteristic equation, we can now construct the general solution of the differential equation. The roots \(r_1 = 3\) and \(r_2 = 5\) tell us the solution to our differential equation will be a combination of exponential functions. Specifically, the general solution to a second-order linear homogeneous differential equation with distinct real roots is given by: \(y = C_1 e^{r_1 x} + C_2 e^{r_2 x}\). Substituting the roots we found, we get: \(y = C_1 e^{3x} + C_2 e^{5x}\). Here, \(C_1\) and \(C_2\) are constants determined by the initial conditions of the problem. This form is important as it shows the solution as a linear combination of exponential functions based on the roots of the characteristic equation.