91Ó°ÊÓ

Undamped motion is a term used to describe systems where there is no energy loss over time. In the context of differential equations, it signifies a motion that is not subject to damping forces, such as friction or air resistance. This results in an oscillation that continues indefinitely with a constant amplitude.

In the exercise, the differential equation given is a model of undamped motion. This is because the equation lacks any terms that would account for damping effects. The equation is:

• \(a y^{\prime \prime}+b y=\cos (\omega t)\)

The absence of a damping term (such as one that would involve a first derivative \(y'\)) indicates that energy is conserved and the system’s oscillations maintain their amplitude over time. Understanding this concept is crucial for solving the exercise, as it sets the groundwork for analyzing the system's behavior and ensuring the solutions reflect the undamped nature.

The characteristic equation is a fundamental concept used to solve linear differential equations, particularly those with constant coefficients. It is derived from the homogeneous part of the differential equation, which is obtained by setting the right-hand side (in this case, \(\cos (\omega t)\)) to zero.

For the given equation \(a y^{\prime \prime} + b y = 0\), simplifying leads to:

• \(y^{\prime \prime} + \frac{b}{a} y = 0\)

Replacing the derivative terms with powers of \(r\) gives the characteristic equation:

• \(r^2 + \frac{b}{a} = 0\)

Solving this quadratic equation provides the roots \(r = \pm i \sqrt{\frac{b}{a}}\), which are imaginary. These roots suggest that the homogeneous solution involves sinusoidal functions - specifically, sines and cosines, indicating oscillatory behavior typical of undamped motion.

In the context of solving non-homogeneous differential equations, finding a particular solution is essential to account for the non-homogeneous part (in this case, \(\cos(\omega t)\)). A particular solution is a specific solution that satisfies the entire non-homogeneous differential equation.

For this exercise, we propose a particular solution of the form:

• \(y_p(t) = A \cos(\omega t) + B \sin(\omega t)\)

This form is chosen because it mirrors the forcing function \(\cos(\omega t)\) on the right-hand side of the equation. Substituting \(y_p(t)\) into the original differential equation and solving for the coefficients \(A\) and \(B\) yields:

• \(A = \frac{1}{b - a \omega^2}\)
• \(B = 0\)

This particular solution ensures the equation is balanced and all terms involving \(\omega t\) are accounted for correctly.

A homogeneous solution, derived from the homogeneous differential equation \(a y^{\prime \prime} + b y = 0\), captures the natural behavior of the system without any external forces (represented by the right-hand side of the original equation). It includes all possible solutions to the homogeneous equation, usually expressed in terms of arbitrary constants.

In this exercise, solving the homogeneous differential equation provides the homogeneous solution:

• \(y_h(t) = C_1 \cos(\omega_b t) + C_2 \sin(\omega_b t)\)

where \(\omega_b = \sqrt{\frac{b}{a}}\). The terms \(C_1\) and \(C_2\) are constants determined by initial conditions and represent the amplitudes of the cosine and sine components of the solution, respectively. These components describe the oscillatory behavior indicative of undamped motion, driven by the imaginary roots of the characteristic equation.