91Ó°ÊÓ

In the context of solving differential equations, the characteristic equation is a core concept. For the given differential equation \[0.75 x'' + \beta x' + 6x = 0,\]we start by assuming a solution of the type\[x = e^{mt}.\]This assumption allows us to transform the original equation into what we call the characteristic equation:\[0.75m^2 + \beta m + 6 = 0.\]This equation is crucial as it transforms the problem from a differential equation into an algebraic one, making it easier to handle. Finding the characteristic equation is a vital step because its roots give us insight into the form of the general solution. It essentially tells us how the solution behaves over time, including whether it oscillates or decays.

Once we have the characteristic equation \[0.75m^2 + \beta m + 6 = 0,\]our next task is to solve it for its roots. This involves applying the quadratic formula:\[m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]where in our specific context, \(a = 0.75\), \(b = \beta\), and \(c = 6\). When you plug these values into the formula, you get:\[m = -\frac{2}{3}\beta \pm \frac{2}{3} \sqrt{\beta^2 - 18}.\]These roots determine the form of our solution. If the term under the square root (\[\beta^2 - 18\]) is positive, the roots are real and distinct. This affects the usage of hyperbolic functions in the solution. Understanding roots is key, as they provide the fundamental building blocks for constructing the full solution.

With our general solution constructed using the roots of the characteristic equation, \[x(t) = e^{-\frac{2}{3}\beta t}\left(c_1 \cosh\left(\frac{2}{3}\sqrt{\beta^2-18} t\right) + c_2 \sinh\left(\frac{2}{3}\sqrt{\beta^2-18} t\right)\right),\]we need specific values to solve for the constants \(c_1\) and \(c_2\). This is where initial conditions come into play. For this problem, we are given:

• \(x(0) = 0\)
• \(x'(0) = -2\)

Plugging \(t = 0\) into the general solution and its derivative gives equations that allow us to isolate \(c_1\) and \(c_2\). Initial conditions are essential because they tailor the general solution to a specific scenario. Without them, the solution remains too vague, providing only a general behavior rather than a precise answer.

By combining the roots of the characteristic equation with our hyperbolic functions, we form the general solution. It captures all possible behaviors of the system described by the differential equation:\[x(t) = e^{-\frac{2}{3}\beta t}\left(c_1 \cosh\left(\frac{2}{3}\sqrt{\beta^2-18} t\right) + c_2 \sinh\left(\frac{2}{3}\sqrt{\beta^2-18} t\right)\right).\]The presence of \(\cosh\) and \(\sinh\) suggests that these functions handle scenarios where the roots are real, which usually leads to exponential growth or decay rather than oscillations. In this way, the general solution offers a comprehensive overview of potential system behaviors before specific initial conditions are applied. By using this framework, we can accommodate a wide range of scenarios just by tweaking \(c_1\) and \(c_2\) based on how the system starts.