91Ó°ÊÓ

In the world of differential equations, encountering complex roots is not uncommon, especially in second-order linear equations. When we solve the characteristic equation like \( m^2 + 4 = 0 \), the result is a pair of complex conjugate roots, \( m_1 = 2i \) and \( m_2 = -2i \). Complex roots occur when the discriminant (the part under the square root in the quadratic formula) is negative, indicating that no real roots exist.

Complex roots are important because they indicate oscillatory solutions, which are common in systems modeling harmonic oscillations, such as in electrical circuits or mechanical vibrations. Instead of exponential growth or decay seen with real roots, the solution will involve sinusoidal functions like sine and cosine. This connection between complex roots and sinusoidal functions is foundational in understanding oscillating systems.

The complementary, or homogeneous, solution of a differential equation involves solving the equation assuming no external force or input. In our case, with complex roots obtained earlier, the complementary solution takes the form of sine and cosine functions. Specifically, the complementary solution derived from the roots \( m_1 = 2i \) and \( m_2 = -2i \) is:

• \( y_c = c_1 \cos(2x) + c_2 \sin(2x) \)

Here, \( c_1 \) and \( c_2 \) are constants determined by initial conditions. These constants allow the solution to be adjusted to fit the particular conditions of a problem, such as initial velocity or position in a mechanical system.

This solution represents the natural behavior of the system without any external influences, showcasing how the system inherently oscillates due to its properties.

To solve a non-homogeneous differential equation, one must find a particular solution that accounts for external forces or inputs. The presence of terms similar to those in the complementary solution often leads to the particular solution being multiplied by \( x \) to distinguish it and avoid redundancy.

In this instance, we preassumed a particular solution with the same frequency components:

• \( y_p = Ax\cos(2x) + Bx\sin(2x) \)

By substituting back into the differential equation, we were able to determine the constants \( A \) and \( B \) through a process called "matching coefficients," leading to:

• \( A = -\frac{3}{4} \)
• \( B = 0 \)

Ultimately, the specific external force described by the differential equation shaped this particular solution:

• \( y_p = -\frac{3}{4}x \cos(2x) \)

Sinusoidal solutions arise when differential equations feature complex roots, as they directly map to the oscillatory nature of sine and cosine waves. These solutions are characterized by frequencies and amplitudes that can describe everything from sound waves to electrical signals.

• The sinusoidal complementary solution is:\( y_c = c_1 \cos(2x) + c_2 \sin(2x) \)
• The sinusoidal particular solution is:\( y_p = -\frac{3}{4}x \cos(2x) \)

When these components are combined, they form a complete solution that accounts for both the system's inherent oscillation and any external forcing terms:

• \( y = y_c + y_p = c_1 \cos(2x) + c_2 \sin(2x) - \frac{3}{4}x \cos(2x) \)

This blending of sinusoidal functions is crucial in understanding and predicting the behavior of systems under various influences, showcasing the elegant harmony between natural oscillations and driven responses.