91Ó°ÊÓ

When dealing with second-order linear homogeneous differential equations like \(y'' - y' + 7y = 0\), the first task is to formulate the characteristic equation. The characteristic equation is crucial because it translates the differential equation into a simpler algebraic equation. By substituting \(y = e^{rt}\), where \(r\) is a constant, into the differential equation, we can derive this characteristic polynomial. Taking derivatives with respect to \(t\), we transform the original differential equation into:

• \(r^2e^{rt} - re^{rt} + 7e^{rt} = 0\)

Factoring out \(e^{rt}\), which is never zero, gives us the characteristic equation \(r^2 - r + 7 = 0\). The beauty of this transformation is that it allows us to find the roots, \(r\), which are instrumental in writing the general solution of the differential equation.

Solving the characteristic equation \(r^2 - r + 7 = 0\) involves finding its roots. Often, we use the quadratic formula:

• \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our case, the coefficients are \(a = 1\), \(b = -1\), and \(c = 7\). By substituting these values, we find:

• \(r = \frac{1 \pm \sqrt{1 - 28}}{2} = \frac{1 \pm \sqrt{-27}}{2}\)

This gives us complex roots \(r_{1,2} = \frac{1}{2} \pm \frac{\sqrt{27}}{2}i\). Complex roots indicate that the solutions will involve sine and cosine terms, reflecting oscillatory behavior within the solution. Recognizing when roots are complex helps predict which trigonometric functions will feature in the general solution.

With the complex roots \(r_{1,2} = \frac{1}{2} \pm \frac{\sqrt{27}}{2}i\), the general solution to the differential equation combines exponential and oscillatory functions. The formula for the general solution of a homogeneous differential equation with complex roots is:

• \(y = e^{\alpha t}(C_1\cos(\beta t) + C_2\sin(\beta t))\)

Here, \(\alpha\) is the real part of the complex roots, and \(\beta\) is the coefficient of the imaginary part. From our roots, we identify \(\alpha = \frac{1}{2}\) and \(\beta = \frac{\sqrt{27}}{2}\). Thus, the general solution of the differential equation is:

• \(y = e^{\frac{t}{2}}(C_1\cos(\frac{\sqrt{27}}{2}t) + C_2\sin(\frac{\sqrt{27}}{2}t))\)

This illustrates how the integration of exponential decay and oscillation constructs the behavior of solutions, characterizing systems that balance these dynamic aspects.