91Ó°ÊÓ

When tackling differential equations, one very useful tool is forming the characteristic equation. This process simplifies the complex task of solving a differential equation by transforming it into an algebra problem. For example, take the differential equation from the exercise:
\(y'' - 4y' + 4y = 0\).
Here, the characteristic equation is constructed by replacing each derivative of \(y\) with \(r^n\), where \(n\) is the order of the derivative, and \(r\) is simply a variable representing the derivative's coefficient. So, in the provided example, we get:
\(r^2 - 4r + 4 = 0\).
This is a straightforward algebraic expression that can be solved for \(r\) to find the solution's nature.

The quadratic formula is a reliable method for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It starts by identifying the coefficients: \(a\), \(b\), and \(c\), which are the numerical factors in front of the terms. Once identified, these coefficients are plugged into the formula
\( r = \frac{-b \bpm \b\b\bsqrt{b^2 - 4ac}}{2a} \).
This formula will provide the roots, or solutions, to the equation, which can be real or complex numbers. By applying this to solve the characteristic equation from our exercise, we determined the roots \(r_1\) and \(r_2\), which are both equal to 2.

After finding the roots of the characteristic equation, we can express the general solution of a linear homogeneous differential equation. The nature of this solution is tightly linked to the roots we obtained. For distinct real roots, the solution is a linear combination of exponential functions. However, for repeated real roots, like in our example with \(r_1 = r_2 = 2\), the solution also involves a polynomial term. Hence, the general solution incorporates both the exponential function and its product with \(x\) to account for the multiplicity of the root:
\[ y = c_1 e^{2x} + c_2 x e^{2x} \].
Here, \(c_1\) and \(c_2\) are constants that you'll determine based on initial conditions if they're provided. This formula encapsulates all possible solutions to the original differential equation.

The superposition principle is a fundamental concept in linear systems such as the differential equations we're studying. This principle states that in any linear system, the net response at a given place and time caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So, when applying superposition to solve linear homogeneous differential equations, one can add the solutions corresponding to each root to find the general solution. This is exactly what we did in the earlier step to achieve the general solution for repeated roots: we combined \(e^{rx}\) and \(x e^{rx}\) due to the double root at \(r=2\). This combination respects the superposition principle by incorporating both forms of the solution that correspond to the repeated root.