91Ó°ÊÓ

The first step in solving the second-order differential equation, \(y^{\prime \prime}+y^{\prime}+y=0\), is to find its characteristic equation. This equation is crucial because it helps us determine the type of roots (real or complex) we have, which directly influences the general solution.
For a general second-order linear homogeneous differential equation with constant coefficients, \(ay^{\prime\prime} + by^{\prime} + cy = 0\), the corresponding characteristic equation is \[ ar^2 + br + c = 0 \]. To obtain it, replace each derivative of \(y\) with \(r\) raised to a power that matches the order of the derivative:
Let's consider our specific example. The differential equation is \( y^{\prime \prime} + y^{\prime} + y = 0 \). Therefore, the characteristic equation will be:
\[ r^2 + r + 1 = 0 \]
This quadratic equation is what we need to solve to move forward.

Next, we solve the characteristic equation obtained: \[ r^2 + r + 1 = 0 \]
We use the quadratic formula for this purpose. The quadratic formula is:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, \(a = 1\), \(b = 1\), and \(c = 1\). Plug these values into the formula:
\[ r = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]
We observe that the discriminant \( \Delta = b^2 - 4ac = 1 - 4 = -3 \) is negative. This indicates that the roots are complex.
Therefore, the roots are:
\[ r_1 = \frac{-1 + \sqrt{3}i}{2} \]
\[ r_2 = \frac{-1 - \sqrt{3}i}{2} \]
Here, \( \alpha = -\frac{1}{2} \) and \( \beta = \frac{\sqrt{3}}{2} \).

Once we have the complex roots, we can write the general solution of the differential equation. For a second-order differential equation with complex roots of the form \(\alpha \pm \beta i\), the solution is expressed as:
\[ y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t)) \]
From our roots, we have \(\alpha = -\frac{1}{2}\) and \(\beta = \frac{\sqrt{3}}{2}\). Substituting these values into the general expression, we get the final solution:
\[ y(t) = e^{-\frac{t}{2}} (C_1 \cos(\frac{\sqrt{3}t}{2}) + C_2 \sin(\frac{\sqrt{3}t}{2})) \]
Understanding how we derive the general solution is crucial. These steps ensure that we correctly apply the initial conditions and solve real-world problems modeled by such differential equations.