91Ó°ÊÓ

To solve second-order linear homogeneous differential equations like \(x^{\rrime} + 4x^{\rpime} + 5x = 0\), we first need to form the characteristic equation. This step is important because it transforms a differential equation into a simpler algebraic equation.

The general form of a second-order linear homogeneous differential equation with constant coefficients is:
\(ax^{\rrime} + bx^{\rpime} + cx = 0\).

To form the characteristic equation, replace the second derivative \(x''\) with \(r^2\), the first derivative \(x'\) with \(r\), and the function \(x\) with 1.

For our given equation \(x^{\rrime} + 4x^{\rpime} + 5x = 0\), the characteristic equation becomes:
\(r^2 + 4r + 5 = 0\).

This characteristic equation is a quadratic equation which we can solve to find the roots.

Solving the characteristic equation \(r^2 + 4r + 5 = 0\) involves using the quadratic formula:

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

Here, \(a = 1\), \(b = 4\), and \(c = 5\). To find the roots, first calculate the discriminant:

\(b^2 - 4ac = 16 - 20 = -4\).

The negative discriminant indicates that the roots are complex. Complex roots have the form:

\(r = \alpha \pm i\beta\).

Using the quadratic formula, we get the complex roots as:
\(r = \frac{-4 \pm \sqrt{-4}}{2} = \frac{-4 \pm 2i}{2} = -2 \pm i\).

Here, \(-2\) is the real part \(\alpha\) and \(1\) is the imaginary part \(\beta\). Complex roots relate to oscillatory solutions in the original differential equation.

The general solution for a differential equation with complex roots is derived from the roots \(r_1 = -2 + i\) and \(r_2 = -2 - i\). When roots are complex, the solution includes exponential functions combined with sine and cosine functions.

Using the formula:
\(x(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t))\),

we can write the solution where \(\alpha\) is the real part and \(\beta\) is the imaginary part. For our equation, \(\alpha = -2\) and \(\beta = 1\).

This gives us the general solution:
\(x(t) = e^{-2t} (C_1 \cos(t) + C_2 \sin(t))\).

Here, \(C_1\) and \(C_2\) are constants determined by initial conditions. This form of the solution signifies a function experiencing exponential decay coupled with oscillations, typical for systems with complex roots.