91Ó°ÊÓ

One of the core concepts in solving second-order differential equations is forming the characteristic equation. This equation is derived by substituting each derivative in the differential equation with powers of an auxiliary variable, typically represented as 'r'. For instance, given the differential equation: \(3 y^{\backprime\backprime} - 2 y^{\backprime} + 10 y = 0\), we substitute to form:
\[3r^2 - 2r + 10 = 0\]
This is a quadratic equation since the highest power of 'r' is 2. To solve this, one must use the quadratic formula which is: \[r = \frac{{-b \backslash pm \backslash sqrt{b^2 - 4ac}}}{2a}\]
Here, you substitute a, b, and c with the coefficients from the characteristic equation (in this case, \(a = 3, b = -2, c = 10\)).

While solving the quadratic characteristic equation, we might encounter complex roots. To identify if the roots are complex, check the discriminant \(b^2 - 4ac\). If it’s negative, the roots are complex. Let's apply this to our equation:
\[3r^2 - 2r + 10 = 0\]
Using the quadratic formula:
\[r = \frac{2 \backslash pm \backslash sqrt{(-2)^2 - 4(3)(10)}}{2(3)} = \frac{2 \backslash pm \backslash sqrt{-116}}{6} = \frac{1 \backslash pm i \backslash sqrt{29}}{3}\]
These roots are complex because of the negative discriminant. It indicates oscillatory solutions and can be written as \(r = \backslash alpha \backslash pm i \backslash beta\). The complex roots are crucial for forming the general solution as they dictate the sinusoidal nature of the solution.

With the complex roots identified, we move on to formulating the general solution of the differential equation. The general solution for complex roots \(r = \backslash alpha \backslash pm i \backslash beta\) is structured as follows:
\[y(t) = e^{\backslash alpha t}\backslash (C_1\backslash cos(\backslash beta t) + C_2\backslash sin(\backslash beta t)\backslash )\]
In our solved example, \(\backslash alpha\) and \(\backslash beta\) are derived from the roots \(\backslashfrac{1 \backslash pm i \backslash sqrt{29}}{3}\), giving us \(\backslash alpha = \backslashfrac{1}{3}\) and \(\backslash beta = \backslashfrac{\backslash sqrt{29}}{3}\). Thus, the general solution is expressed as:
\[y(t) = e^{\backslashfrac{1}{3} t}\backslash (C_1 \backslash cos(\backslashfrac{\backslash sqrt{29}}{3} t) + C_2 \backslash sin(\backslashfrac{\backslash sqrt{29}}{3} t)\backslash )\]
Here, \(C_1\) and \(C_2\) are arbitrary constants and are determined by initial conditions.