91Ó°ÊÓ

When dealing with differential equations, particularly those involving oscillatory behavior, we often encounter the term "steady periodic solution." Essentially, this solution represents the behavior of the system once all transient effects have dissipated. It's the long-term behavior of the system, driven entirely by the external forces or inputs, such as sine or cosine functions. In our exercise, we're dealing with a differential equation subject to a sinusoidal driving force.The steady periodic solution for a differential equation like the one in our exercise is derived from attempting solutions of the form \( x_{sp}(t) = C \cos(\omega t - \alpha) \), where \( \omega \) is the frequency of the external force, \( C \) is the amplitude, and \( \alpha \) is the phase shift. By substituting this into the equation and matching coefficients, we find the values for \( C \) and \( \alpha \) that allow the solution to satisfy the original differential equation.

• The amplitude \( C \) determines the size of the oscillation.
• The phase shift \( \alpha \) adjusts where the wave starts along the time axis.

For our problem, we found \( x_{sp}(t) = \frac{10}{39} \cos(5t - \frac{\pi}{2}) \). This shows us that after the initial effects have died away, the system will oscillate at this amplitude and phase.

Transient solutions describe how the system behaves from the moment it starts until just before it settles into a steady state. This part of the solution stems from the initial conditions provided in the problem.The transient solution generally consists of terms from the homogeneous solution \( x_h(t) \), due to the nature of the given differential equation. These terms eventually fade away over time because they are typically exponential decay functions. The speed at which these terms diminish is determined by the decay rate in the exponential factor.In the given problem, the transient solution translates to \( x_{tr}(t) = e^{-3t}(-\frac{10}{39} \cos 2t) \). This function addresses the part of the system's response caused by initial displacements or velocities. With time, the influence of these initial conditions weakens, as evidenced by the exponential decay term \( e^{-3t} \), which indicates the rate of decay in this context.

The characteristic equation is an essential tool for anyone tackling differential equations. It emerges when we look at the homogeneous part of the differential equation, which lacks external forces or inputs. We often search for solutions of the form \( e^{rt} \), leading to the characterizing equation governing the system's intrinsic behavior.In the context of our problem, the homogeneous differential equation is \( x'' + 6x' + 13x = 0 \). To find the solutions to this, we derive the characteristic equation, which is a quadratic equation in terms of \( r \): \( r^2 + 6r + 13 = 0 \).

• This equation determines the nature of the system’s response without external influence.
• By finding the roots of this characteristic polynomial, we can determine the system's natural frequencies and damping.

Using the quadratic formula, we find the roots \( r = -3 \pm 2i \). These roots indicate an oscillatory solution with an exponential decay factor, typifying systems with damping, such as underdamped harmonic oscillators. The roots contribute directly to the homogeneous solution \( x_h(t) \), which forms the foundation for the transient part of the response.