Question

Suppose that you have found a particular solution \(x_{\mathrm{p}}(t)\) of the inhomogeneous equation (5.48) for a driven damped oscillator, so that \(D x_{\mathrm{p}}=f\) in the operator notation of \((5.49) .\) Suppose also that \(x(t)\) is any other solution, so that \(D x=f .\) Prove that the difference \(x-x_{\mathrm{p}}\) must satisfy the corresponding homogeneous equation, \(D\left(x-x_{\mathrm{p}}\right)=0 .\) This is an alternative proof that any solution \(x\) of the inhomogeneous equation can be written as the sum of your particular solution plus a homogeneous solution; that is, \(x=x_{\mathrm{p}}+x_{\mathrm{h}}\).

Short answer

The difference \(x - x_{\mathrm{p}}\) satisfies the homogeneous equation, proving \(x = x_{\mathrm{p}} + x_{\mathrm{h}}\).

Step-by-step solution

Define the Given Functions

We are given that \(x_{\mathrm{p}}(t)\) is a particular solution of the inhomogeneous equation \(D x_{\mathrm{p}} = f\), and \(x(t)\) is any solution of the same equation \(D x = f\). Thus, both satisfy the inhomogeneous differential equation described by operator notation.

Express the Difference Function

We are asked to consider the function \(y(t) = x(t) - x_{\mathrm{p}}(t)\). We hypothesize that this function satisfies the homogeneous equation. The task is to prove that \(D(y) = 0\).

Substitute into the Operator

Substitute \(y(t) = x(t) - x_{\mathrm{p}}(t)\) into the differential operator \(D\): \[ D(y) = D(x - x_{\mathrm{p}}) = D(x) - D(x_{\mathrm{p}}). \]

Apply Given Differential Equations

Since \(D(x) = f\) and \(D(x_{\mathrm{p}}) = f\), substitute these into the equation from the previous step:\[ D(y) = f - f = 0. \]

Conclusion of the Proof

Since \(D(y) = 0\), the function \(y(t)\) satisfies the homogeneous equation. Consequently, any solution \(x(t)\) of the inhomogeneous equation can be expressed as the sum:\[ x(t) = x_{\mathrm{p}}(t) + x_{\mathrm{h}}(t), \]where \(x_{\mathrm{h}}(t)\) is a solution to the homogeneous equation \(D(x_{\mathrm{h}}) = 0\).

Key concepts

Driven Damped Oscillator

A driven damped oscillator is a system where a damping force and an external driving force act on an oscillating object. This is common in many real-world scenarios, such as in car suspension systems or pendulums affected by air resistance and periodic pushes.

In physics terms, the damping force usually takes away energy from the system, causing the oscillating motion to gradually decrease over time. The driving force, on the other hand, adds energy to the system, which can sustain or even increase the amplitude of the oscillations.

• Damping force typically has a negative impact on motion, slowing it down.
• Driving force is often external and periodic, designed to influence the oscillator.

Understanding the balance of these forces is crucial in predicting how the system will behave over time, often modeled with differential equations to quantify the effect mathematically.

Inhomogeneous Equation

An inhomogeneous equation in the context of differential equations involves a function on the right-hand side. This makes the equation non-uniform across the domain of solutions.

For a driven damped oscillator described earlier, this function is the external driving force, a term added to the differential equation of the system, represented as follows: \[ Dx = f, \] where \( f \) is a non-zero function.

• The presence of \( f \) introduces complexity, as solutions can't simply return to equilibrium without external influence.
• Inhomogeneous equations require different techniques to solve, including finding a particular solution that satisfies the equation.

Repeatedly encountering such equations suggests systems affected by external factors, including electrical circuits, mechanical systems, and thermal processes.

Homogeneous Equation

A homogeneous equation is a special type of differential equation where the right-hand side is zero. This signifies that there are no external forces acting on the system.

In the context of our exercise, it is written as:\[ D(x-x_p) = 0. \] Understanding these allows us to focus solely on the intrinsic properties of the system itself.

• Solutions to homogeneous equations only depend on initial conditions and natural tendencies of the system.
• In mechanical terms, it means systems return to their base state without external disruption.

The beauty of homogeneous solutions often lies in their simplicity and purity, revealing fundamental characteristics of the oscillators or other systems described.

Particular Solution

The particular solution is a crucial component in solving inhomogeneous differential equations. It's a specific solution that satisfies the non-zero right side of the equation.

For the driven damped oscillator, it is the particular solution \( x_p(t) \) that accommodates the external force \( f \) applied:
\[ D x_p = f. \]

• Finding this particular solution often involves making educated guesses or using known problem-solving methods like the undetermined coefficients or variation of parameters.
• It represents the specific way the system responds to external influences, considering all forces applied.

Adding this to the general solution of the homogeneous equation gives us the overall response of the system.

Operator Notation

Operator notation is a powerful method of representing differential equations concisely. It employs mathematical operators to manage derivatives without expanding the full form each time.

In our scenario, the operator \( D \) is used to signify the differentiation process. We see it applied as:\[ D x = f \] for inhomogeneous equations, and \[ D(x-x_p) = 0 \] for homogeneous equations.

• This method streamlines complex differential equations, making solutions more accessible and manageable.
• It allows focusing on solving techniques instead of repeatedly dealing with the difference quotient.

Acknowledging notation differences can help in understanding the full spectrum of approaches needed for solving these equations efficiently.