91Ó°ÊÓ

The characteristic equation is vital in solving second-order linear homogeneous differential equations. Imagine it as a bridge connecting the differential equation to its solution. For a differential equation of the form \(ay'' + by' + cy = 0\), the characteristic equation is \(ar^2 + br + c = 0\). This quadratic equation helps determine the nature of the solutions.
In our exercise, the differential equation is \(3y'' + y = 0\), where \(a = 3\), \(b = 0\), and \(c = 1\). Therefore, the characteristic equation simplifies to:

• \(3r^2 + 1 = 0\)

This equation is essential because solving it gives us the roots that indicate the type of solutions we should expect.
Remember, the characteristic equation simplifies our task by reducing a differential equation problem to a polynomial problem, which is often more straightforward to solve.

When solving the characteristic equation \(3r^2 + 1 = 0\), we notice that it involves complex numbers. This happens because the equation results in a negative number under the square root when solving for \(r\):

• Set the equation \(3r^2 = -1\).
• Divide both sides by 3: \(r^2 = -\frac{1}{3}\).
• The roots \(r\) are found by taking the square root, resulting in \(r = \pm i\sqrt{\frac{1}{3}}\).

These roots, expressed as \(\alpha \pm i\beta\), indicate that our solution involves complex roots, where:\

• \(\alpha = 0\) is the real part.
• \(\beta = \sqrt{\frac{1}{3}}\) is the imaginary part.

Complex roots suggest solutions involving trigonometric functions, namely cosine and sine. This occurs because complex numbers allow for oscillatory solutions, common in systems like springs and circuits.

Upon identifying complex roots, we proceed to form the general solution using a specific formula. For differential equations with roots \( \alpha \pm i\beta \), the general solution is:
\[ y(t) = e^{\alpha t}(C_1 \cos \beta t + C_2 \sin \beta t) \]
In our case, \(\alpha = 0\) and \(\beta = \sqrt{\frac{1}{3}}\), simplifying our formula to:

• Remove \(e^{0t}\) since it equals 1, giving:
• \(y(t) = C_1 \cos(\sqrt{\frac{1}{3}}t) + C_2 \sin(\sqrt{\frac{1}{3}}t)\)

This solution represents all possible functions that satisfy the original differential equation. The constants \(C_1\) and \(C_2\) are determined by initial or boundary conditions, allowing the solution to fit specific physical or real-world constraints.
The beauty of this general solution lies in its flexibility. It can model various systems that behave in an oscillatory manner due to their complex roots.