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Chapter 4: Problem 3

Determine the equation of motion for an undamped system at resonance governed by $$ \begin{array}{l}{\frac{d^{2} y}{d t^{2}}+9 y=2 \cos 3 t} \\ {y(0)=1, \quad y^{\prime}(0)=0}\end{array} $$ Sketch the solution.

Short Answer

Expert verified
The solution to the ODE is \(y(t) = \cos(3t) - \frac{2}{27} \sin(3t) - \frac{2}{9} \sin(3t)\), and when plotted, the solution represents a resonate oscillator, start at 1 at t=0, and oscillates with an increasing amplitude due to the resonance effect with a frequency of 3.

Step by step solution

01

Identify the Homogeneous Solution

The associated homogeneous equation is \( \frac{d^2 y}{dt^2} + 9y = 0 \). Its roots are \( \pm 3i \), corresponding to the general solution \( y_h(t) = A \cos(3t) + B \sin(3t) \).
02

Find the Particular Solution

As our forcing function is \(2 \cos(3t)\), we propose a particular solution of the form \(y_p(t) = C \cos(3t) + D \sin(3t)\). Substituting this into the original ODE, and comparing coefficients, gives the system of equations \(C = 0\) and \(-9D = 2\), leading to \(D = -2/9\). Thus, the particular solution is \(y_p(t) = -\frac{2}{9} \sin(3t)\).
03

Combine Homogeneous and Particular Solutions

The general solution of the ODE is the sum of the homogeneous solution and the particular solution, which gives \(y(t) = A \cos(3t) + B \sin(3t) - \frac{2}{9} \sin(3t)\).
04

Apply Initial Conditions

From initial conditions \(y(0) = 1\) and \(y'(0) = 0\), we derive \(A = 1\), and \(-3B - \frac{2}{9}\cdot3 = 0\), so \(B = -\frac{2}{27}\). Thus, the solution that obey the given initial conditions is \(y(t) = \cos(3t) - \frac{2}{27} \sin(3t) - \frac{2}{9} \sin(3t)\).
05

Sketching the Solution

To sketch, plot the function \(y(t) = \cos(3t) - \frac{2}{27} \sin(3t) - \frac{2}{9} \sin(3t)\), remembering that the solution represents simple harmonic motion at resonance. At t=0, we start at 1, and the solution oscillates with increasing amplitude due to the resonance effect, with a frequency of 3 (as seen in the arguments of sin and cos).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Homogeneous Solution
In the study of differential equations, particularly with resonance, finding the homogeneous solution forms the foundational step in solving these equations. A homogeneous differential equation is one where the right-hand side (forcing function) is zero. For the given equation \( \frac{d^2 y}{dt^2} + 9y = 0 \), we have a second-order linear homogeneous differential equation.

To solve it, we substitute the general solution form, assuming solutions of the type \( y_h(t) = e^{rt} \). By substituting this into the differential equation, we find the characteristic equation:
\[ r^2 + 9 = 0 \]
Simplifying, the roots are \( r = \pm 3i \), which indicates that the solution involves complex numbers. This result suggests the solution will be sinusoidal, specifically:
  • \( A \cos(3t) + B \sin(3t) \)
This solution highlights the simple harmonic motion component of the system, characterized by the natural frequency \( 3 \).
The homogeneous solution is essential as it represents the system's natural response without external forcing.
Particular Solution
The particular solution deals specifically with the non-homogeneous aspect of a differential equation, essentially accounting for the external forcing function. In our equation, \( 2 \cos(3t) \) serves as this external force.

To find a particular solution, we propose a similar form to the forcing function, \( y_p(t) = C \cos(3t) + D \sin(3t) \). Substituting this form back into the original differential equation allows us to determine the coefficients through comparison:
  • Substituting, we find \( C = 0 \).
  • For \( D \), the equation simplifies to \( -9D = 2 \), thus \( D = -\frac{2}{9} \).
The particular solution is hence:
\[ y_p(t) = -\frac{2}{9} \sin(3t) \]
This solution is crucial as it reflects how external forces influence the system, leading to resonance, particularly when the frequency of \( \cos(3t) \) matches the system's natural frequency.
Initial Conditions
Initial conditions are essential in determining a unique solution to a differential equation, especially in physical scenarios like motion. For this problem, the initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \) are given.

Once the general solution \( y(t) = A \cos(3t) + (B - \frac{2}{9})\sin(3t) \) is formulated, we apply initial conditions to resolve \( A \) and \( B \).
  • Setting \( y(0) = 1 \), directly gives \( A = 1 \).
  • Your next move is based on speed at zero: \( y'(0) = 0 \), which works out to \(-3B - \frac{6}{27} = 0 \), solving for \( B = -\frac{2}{27} \).
The initial conditions crucially tie the mathematical solution back to the physical context, ensuring it reflects the real world, particularly the system's beginning state.
Simple Harmonic Motion
Simple harmonic motion (SHM) is an oscillatory motion under a restoring force proportional to displacement. The system described by the equation \( \frac{d^2 y}{dt^2} + 9y = 2 \cos(3t) \) exemplifies this, possessing natural oscillation, influenced by external forces.

In this case, the system is initially at rest and then partakes in harmonic motion, characterized by the terms \( \cos(3t) \) and \( \sin(3t) \).
  • SHM involves periodic functions, revealing how positions change over equal time intervals.
  • The magnitude of these oscillations can increase significantly at resonance, as shown here by the growing amplitude over time.
Resonance occurs when the system's natural frequency coincides with the external force frequency. This situation amplifies the motion, making energy transfer efficient. The physical manifestation of the equation shows this increasing amplitude and frequency of 3 rad/s, indicating real-world phenomena like a swinging pendulum or vibrating strings.

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Most popular questions from this chapter

$$z^{\prime \prime}+z^{\prime}-z=0$$

Solve the initial value problem: $$\begin{array}{ll}{y^{\prime \prime \prime}-y^{\prime}=0 ;} & {y(0)=2} \\\ {y^{\prime}(0)=3,} & {y^{\prime \prime}(0)=-1}\end{array}$$

Boundary Value Problems. When the values of a solution to a differential equation are specified at two different points, these conditions are called boundary conditions. (In contrast, initial conditions specify the values of a function and its derivative at the same point.) The purpose of this exercise is to show that for boundary value problems there is no existence-uniqueness theorem that is analogous to Theorem 1. Given that every solution to $$\quad y^{\prime \prime}+y=0$$ is of the form $$y(t)=c_{1} \cos t+c_{2} \sin t$$, where $$c_{1}$$ and $$c_{2}$$ are arbitrary constants, show that (a) There is a unique solution to (17) that satisfies the boundary conditions $$y(0)=2 \text { and } y(\pi / 2)=0$$. (b) There is no solution to (17) that satisfies $$y(0)=2$$ and $$y(\pi)=0$$. (c) There are infinitely many solutions to (17) that satisfy $$y(0)=2$$ and $$y(\pi)=-2$$.

$$y^{\prime \prime \prime}-y^{\prime \prime}+y=\sin t$$

Consider the linear equation \(t^{2} y^{\prime \prime}-3 t y^{\prime}+3 y=0,\) for \(-\infty\) \(<\) \(t\) \(<\) \(\infty\) (a) Verify that \(y_{1}(t) :=t\) and \(y_{2}(t) :=t^{3}\) are two solutions to \((21)\) on \((-\infty, \infty) .\) Furthermore, show that \(y_{1}\left(t_{0}\right) y_{2}^{\prime}\left(t_{0}\right) y_{1}^{\prime}\left(t_{0}\right) y_{2}\left(t_{0}\right) \neq 0\) for \(t_{0}=1 .\) (b) Prove that \(y_{1}(t)\) and \(y_{2}(t)\) are linearly independent on \((-\infty, \infty) .\) (c) Verify that the function \(y_{3}(t) :=|t|^{3}\) is also a solution to \((21)\) on \((-\infty, \infty)\) (d) Prove that there is \(n o\) choice of constants \(c_{1}, c_{2}\) such that \(y_{3}(t)=c_{1} y_{1}(t)+c_{2} y_{2}(t)\) for all \(t\) in \((-\infty, \infty)\) . [Hint: Argue that the contrary assumption leads to a contradiction.] (e) From parts \((\mathrm{c})\) and \((\mathrm{d}),\) we see that there is atleast one solution to \((21)\) on \((-\infty, \infty)\) that is not expressible as a linear combination of the solutions \(y_{1}(t), y_{2}(t) .\) Does this provide a counterexample to the theory in this section? Explain.
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