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When we deal with second-order linear homogeneous differential equations like \(y'' - 2y' - 3y = 0\), the characteristic equation plays a central role. This equation helps us transform a complex differential problem into an easier algebraic problem. We find the characteristic equation by substituting the derivatives in the differential equation with powers of \(r\):

• Replace \(y''\) with \(r^2\)
• Replace \(y'\) with \(r\)
• Replace \(y\) with 1

This results in a characteristic equation such as \(r^2 - 2r - 3 = 0\). The primary aim is to find the roots of this equation, which reveal critical information about the nature of the solution to the original differential equation. Each type of root—real distinct, real repeated, or complex—leads to a different form of solution.

The quadratic formula is a powerful tool that solves any quadratic equation of the form \(ax^2 + bx + c = 0\). In our case, the characteristic equation is \(r^2 - 2r - 3 = 0\). Here, we identify the coefficients as \(a = 1\), \(b = -2\), and \(c = -3\). The quadratic formula is: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Applying the formula, we replace and calculate:

• \(-b = 2\)
• \(b^2 - 4ac = 4 + 12 = 16\)
• The square root of 16 is 4

So the solution is: \[ r = \frac{2 \pm 4}{2} \]This simplifies to \(r_1 = 3\) and \(r_2 = -1\), providing two distinct real roots. These results are crucial for writing the general solution of the differential equation.

Once the roots of the characteristic equation are known, we can construct the general solution to the differential equation. For equations with distinct real roots \(r_1\) and \(r_2\), the general solution takes the form:\[ y(x) = C_1e^{r_1x} + C_2e^{r_2x} \]Here, \(C_1\) and \(C_2\) are arbitrary constants, determined by the initial or boundary conditions of the specific problem you are working on. For our given equation, the roots we found are \(r_1 = 3\) and \(r_2 = -1\). Substituting these values, we arrive at:\[ y(x) = C_1e^{3x} + C_2e^{-x} \]This expression is the general solution, representing a family of functions that satisfy the original differential equation. Each specific solution within this family can be determined through additional information, such as initial conditions.