91Ó°ÊÓ

In differential equations, identifying zero points of functions like cosine is crucial in solving these equations effectively. Let's focus on the cosine function, \(\cos x\). The cosine function has zero values at specific points, which are essential when the function acts as a coefficient of the highest derivative in a differential equation.

The cosine function specifically becomes zero at odd integer multiples of \(\frac{\pi}{2}\). Mathematically, these points can be represented by the expression: \(x = (2n+1)\frac{\pi}{2}\), where \(n\) is any integer. This means the function is zero at such points as \(-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \) etc. Understanding these zero points helps us identify singularities in differential equations where solutions might be undefined or require special attention.

Singularities are locations where a differential equation's behavior becomes problematic or discontinuous. Recognizing singular points is a fundamental part of solving differential equations as they often indicate positions where the solution could become infinite, undefined, or complex to handle.

In our example, the singular points arise when \(\cos x\) — the coefficient of the highest derivative, \(y^{\prime\prime}\) — equals zero. At these points, the differential equation \( (\cos x) y^{\prime \prime} + y^{\prime} + 5y = 0 \) can lose its predictability and require alternative methods for solution. By analyzing where \(\cos x\) equals zero, that is, at odd integer multiples of \(\frac{\pi}{2}\), we pinpoint the singularities of our equation. This form of identification is instrumental in understanding the complexity or changes in behavior of the solutions around those points.

Once singular points are identified, calculating their distance to a specific point of interest on the function’s domain provides valuable insights into the solution’s behavior. This distance calculation aids in recognizing how close a point lies to where a solution might become undefined.

For instance, if \(x = 1\) is our point of interest in the example equation, we can find the nearest singular point, which is the odd multiple of \(\frac{\pi}{2}\) close to 1. The options nearby are \(\pm \frac{\pi}{2}\), with \(\frac{\pi}{2}\) (approximately 1.57) being closer. Thus, we determine the distance from 1 to this point as \(\frac{\pi}{2} - 1\).

Conducting a simple computation, where \(\pi/2\) approximates 1.57, reveals that the exact distance from point 1 to the nearest singular point \(\frac{\pi}{2}\) is \(0.57\). Such calculations help understand the sensitivity of solutions and any potential numerical approximation or stability issues that might occur at or near singular points.