Problem 12 - Differential Equations and Linear Algebra

Question

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. \(\frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=F_{0} \cos \omega t,\) where \(\omega_{0}, \omega\) are positive con- stants, and \(F_{0}\) is an arbitrary constant. You will need to consider the cases \(\omega \neq \omega_{0}\) and \(\omega=\omega_{0}\) separately.

Step-by-Step Solution

Answer

For the given differential equation, we have two cases for the particular solution: 1. \(\omega \neq \omega_{0}\): \(y_p(t) = \frac{F_0}{2(\omega_{0}^2 - \omega^2)}\cos(\omega t)\) 2. \(\omega = \omega_{0}\): \(y_p(t) = t\left(\frac{F_0}{2(1+\omega_0)} \cos(\omega_0 t)+ \frac{F_0}{2\omega_0} \sin(\omega_0 t)\right)\)

Step 1: Case 1: \(\omega \neq \omega_{0}\)

Let's consider the case \(\omega \neq \omega_{0}\). We will use a trial solution of the form: \[y_p(t) = A e^{i\omega t}\] Now, let's differentiate this function twice and substitute it into the given equation to obtain \(A\).

Step 2: Differentiate the trial solution

First derivative of \(y_p(t)\): \[\frac{dy_p(t)}{dt} = i \omega A e^{i\omega t}\] Second derivative of \(y_p(t)\): \[\frac{d^2y_p(t)}{dt^2} = -\omega^2 A e^{i\omega t}\]

Step 3: Substitute and solve for A

Now, substitute these into the given differential equation: \[-\omega^2 A e^{i\omega t} + \omega_{0}^2 A e^{i\omega t} = F_0 \cos(\omega t)\] Divide both sides by \(e^{i\omega t}\): \[-\omega^2 A + \omega_{0}^2 A = F_0 \cos(\omega t)e^{-i\omega t}\] We know that \(\cos(\omega t) = \frac{1}{2}(e^{i\omega t}+e^{-i\omega t})\), so we can rewrite the equation as: \[-\omega^2 A + \omega_{0}^2 A = \frac{F_0}{2}(1 + e^{-2i\omega t}) \Rightarrow A= \frac{F_0}{2(\omega_{0}^2 - \omega^2)}\] Now plug this value back into our trial solution \(y_p(t)\): \[y_p(t) = \frac{F_0}{2(\omega_{0}^2 - \omega^2)} e^{i\omega t}\]

Step 4: Convert back to real form

Using Euler's formula, we can express this complex-valued function back in real form: \[y_p(t) = \frac{F_0}{2(\omega_{0}^2 - \omega^2)} (\cos(\omega t) + i\sin(\omega t))\] As we are looking for a real-valued function, we only need the cosine part: \[y_p(t) = \frac{F_0}{2(\omega_{0}^2 - \omega^2)}\cos(\omega t)\] So, for the case \(\omega \neq \omega_{0}\), the particular solution is: \[y_p(t) = \frac{F_0}{2(\omega_{0}^2 - \omega^2)}\cos(\omega t)\]

Step 5: Case 2: \(\omega = \omega_{0}\)

Now let's consider the case \(\omega = \omega_{0}\). In this case, we will use a trial solution of the form: \[y_p(t) = t(A\cos(\omega_{0} t)+B\sin(\omega_{0}t))\]

Step 6: Differentiate the trial solution

First, let's compute the first and second derivative of the trial solution: First derivative of \(y_p(t)\): \[\frac{dy_p(t)}{dt} = A\cos(\omega_0 t) + B\sin(\omega_0 t) - \omega_0 t (A\sin(\omega_0 t) - B\cos(\omega_0 t))\] Second derivative of \(y_p(t)\): \[\frac{d^2y_p(t)}{dt^2} = -2 \omega_0 A\sin(\omega_0 t) + 2 \omega_0 B\cos(\omega_0 t) - \omega_0^2 t (A\cos(\omega_0 t)+B\sin(\omega_0 t))\]

Step 7: Substitute and solve for A and B

Now, substitute the second derivative of the trial solution into the given differential equation: \[-2 \omega_0^2 F_0\sin(\omega_0 t) + 2 \omega_0^2 F_0\cos(\omega_0 t) - \omega_0^2 F_0\sin(\omega_0 t) = F_0\cos(\omega_0 t)\] Notice that: \[\begin{cases} -2\omega_0 A + A = \frac{1}{2}F_0 \\ \omega_0 B = \frac{1}{2}F_0 \end{cases}\] Solving the system, we get: \[\begin{cases} A = \frac{F_0}{2(1+\omega_0)} \\ B = \frac{F_0}{2\omega_0} \end{cases}\] Now, substitute these values back into the trial solution \(y_p(t)\): \[y_p(t) = t\left(\frac{F_0}{2(1+\omega_0)} \cos(\omega_0 t)+ \frac{F_0}{2\omega_0} \sin(\omega_0 t)\right)\] So, for the case \(\omega = \omega_{0}\), the particular solution is: \[y_p(t) = t\left(\frac{F_0}{2(1+\omega_0)} \cos(\omega_0 t)+ \frac{F_0}{2\omega_0} \sin(\omega_0 t)\right)\] In conclusion, for the given second-order linear differential equation, we obtained the following particular solutions: 1. Case 1 (\(\omega \neq \omega_{0}\)): \(y_p(t) = \frac{F_0}{2(\omega_{0}^2 - \omega^2)}\cos(\omega t)\) 2. Case 2 (\(\omega = \omega_{0}\)): \(y_p(t) = t\left(\frac{F_0}{2(1+\omega_0)} \cos(\omega_0 t)+ \frac{F_0}{2\omega_0} \sin(\omega_0 t)\right)\)

Key Concepts

Particular Solution

In the realm of differential equations, finding a particular solution is a critical step to solving non-homogeneous differential equations. The particular solution represents a specific solution that satisfies the non-homogeneous equation.
It is often denoted as \( y_p(t) \), and it allows us to express the overall solution as the sum of this particular solution and the general solution of the associated homogeneous equation.

• For the problem given, the particular solution helps us to respond to the constant external force expressed as \( F_0 \cos(\omega t) \).
• The differential equation \( \frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=F_{0} \cos \omega t \) describes how the system behaves in response to this input.

To find \( y_p(t) \), different approaches such as the method of undetermined coefficients or variation of parameters can be employed. In this particular case, using a complex-valued trial solution for simplicity often captures the oscillatory nature of the particular solution effectively.

Complex-Valued Trial Solution

Employing a complex-valued trial solution is a mathematically savvy way to unravel periodic solutions in differential equations. The use of complex numbers simplifies calculations, especially when dealing with trigonometric functions.

• The trial solution \( y_p(t) = A e^{i\omega t} \) leverages the properties of exponential functions, which are easier to manipulate than trigonometric expressions.
• After finding \( A \), the complex solution is ultimately converted back to its real counterpart.

This process hinges on Euler's formula: \( e^{i\omega t} = \cos(\omega t) + i\sin(\omega t) \). This conversion is crucial because differential equations describe real-world phenomena and thus require real solutions.

Harmonic Oscillator

The concept of a harmonic oscillator is prevalent in physics, particularly when modeling systems that exhibit oscillatory behavior.

• In the given differential equation, \( \frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=F_{0} \cos \omega t \), the term \( \omega_{0}^{2} y \) denotes the restoring force typical of harmonic systems.
• This equation reflects a system that naturally oscillates at a frequency \( \omega_0 \) when disturbed.

The fundamental characteristic of a harmonic oscillator lies in its proportional restoring force: the further you displace it from equilibrium, the greater the force pulling it back.
Understanding how this principle applies to various systems – like springs and pendulums – provides deeper comprehension of oscillatory motion in both natural and engineered systems.

Resonance

Resonance is a fascinating and sometimes dangerous phenomenon where a system's oscillations are amplified when the frequency of external forces matches the system's natural frequency. In the context of our differential equation, resonance occurs when \( \omega = \omega_0 \).

• At resonance, systems can experience drastic increases in amplitude, leading to large oscillations.
• To solve for a particular solution when \( \omega = \omega_0 \), we use a modified trial solution: \( y_p(t) = t(A\cos(\omega_{0} t)+B\sin(\omega_{0}t)) \).

This adjustment is necessary because directly matching frequencies causes division by zero in the conventional approach.
Managing resonance, whether harnessing it as in musical instruments or mitigating it in structural engineering, is central to both exploiting and protecting against natural frequency phenomena.