91Ó°ÊÓ

When dealing with second order differential equations, understanding how to derive the characteristic equation is essential. The characteristic equation allows us to transform the differential equation into a simpler form that we can solve using algebraic methods.
For the given differential equation \(y'' + 2y' + 5y = 0\), we create a characteristic equation by replacing \(y\) with \(e^{mx}\), which leads to \(m^2 + 2m + 5 = 0\). Here, \(m\) represents the roots of the characteristic equation.
The characteristic equation is essentially a polynomial representation of the differential equation. By solving this, we can identify the roots that will help us form the solution to the original problem.

Sometimes, the roots of the characteristic equation are complex rather than real. In our example, after placing the given values \(a=1\), \(b=2\), and \(c=5\) into the quadratic formula \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we end up with complex roots \(m = -1 \pm 2i\).
Complex roots occur when the discriminant \((b^2 - 4ac)\) is negative, leading to a square root of a negative number. These roots are expressed in the form \(a \pm bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
The presence of complex roots indicates that the solution to the differential equation involves oscillatory components, as represented by cosine and sine terms in the general solution.

The general solution of a second order differential equation provides a formula that incorporates all possible solutions. It expresses the solution in terms of arbitrary constants, accounting for the initial conditions or any additional constraints provided.
In our case, where the characteristic equation yields complex roots \(m = -1 \pm 2i\), the general solution takes the form:

• \(y(x) = e^{ax}(C_1 \cos(bx) + C_2 \sin(bx))\)

Here, \(a = -1\) and \(b = 2\), resulting in:

• \(y(x) = e^{-x}(C_1 \cos(2x) + C_2 \sin(2x))\)

The constants \(C_1\) and \(C_2\) are determined based on initial conditions or constraints.
This unique form shows the relationship between exponential decay \(e^{-x}\) and oscillations \(\cos(2x)\) and \(\sin(2x)\) in the behavior of the solution.