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When dealing with second-order homogeneous linear differential equations like \(y'' - 2y' + 4y = 0\), a key step is to form the characteristic equation. This is an algebraic equation derived directly from the differential equation itself. The standard form for a second-order equation is \(ay'' + by' + cy = 0\).The characteristic equation corresponding to this is \(ar^2 + br + c = 0\). By substituting the coefficients of the differential equation into this standard form, you replace the derivatives with a polynomial in \(r\), which models the behavior of the solutions.

• For our example: \(a = 1\), \(b = -2\), and \(c = 4\).
• Thus, the characteristic equation becomes \(r^2 - 2r + 4 = 0\).

Solving this characteristic equation is the next step to finding the behavior of the solutions of the differential equation.

The roots of the characteristic equation can tell us a lot about the solutions of the original differential equation. Complex roots arise when the discriminant \(b^2 - 4ac\) is negative, meaning that the square root of a negative number is involved. This gives rise to imaginary numbers.In the characteristic equation \(r^2 - 2r + 4 = 0\):

• The discriminant, \(b^2 - 4ac\), is calculated as \((-2)^2 - 4 \times 1 \times 4 = -12\) which is negative.
• Therefore, our roots are complex: \(r = \frac{2 \pm \sqrt{-12}}{2} = 1 \pm i\sqrt{3}\).

These complex roots indicate an oscillatory component in the solution, influenced by both exponentials and trigonometric functions due to their imaginary parts.

Once you have the complex roots of the characteristic equation, you can derive the general solution to the differential equation. For complex roots of the form \(r = \alpha \pm i\beta\), the general solution is expressed using exponential and trigonometric functions:\[ y(x) = e^{\alpha x}\left(C_1 \cos(\beta x) + C_2 \sin(\beta x)\right) \]For the given roots \(1 \pm i\sqrt{3}\), we have:

• \(\alpha = 1\)
• \(\beta = \sqrt{3}\)

Hence, the solution becomes:\[ y(x) = e^{x} \left(C_1 \cos(\sqrt{3}x) + C_2 \sin(\sqrt{3}x)\right) \]This formula represents a family of solutions, where \(C_1\) and \(C_2\) are constants determined by boundary conditions or initial values. They encompass the full range of behaviors the system can exhibit depending on how it's "started" or constrained. The beauty of this solution lies in its ability to capture both the growing/decaying term \(e^{x}\) and the oscillatory behavior \(\cos(\sqrt{3}x)\) and \(\sin(\sqrt{3}x)\).