Question

Show that the general solution of \(\frac{d^{2} w}{d t^{2}}+\frac{e^{2 t}-v^{2}}{t^{4}} w=0\) is \(w=\) \(t\left[A J_{v}\left(e^{1 / t}\right)+B Y_{v}\left(e^{1 / t}\right)\right]\).

Short answer

The given differential equation transforms into Bessel's differential equation under the change of variable \(x = e^{1/t}\). The general solution, thus, is \(w=t\left[A J_{v}\left(e^{1 / t}\right)+B Y_{v}\left(e^{1 / t}\right)\right]\).

Step-by-step solution

Recognize the form of equation

Identify that the given equation \(\frac{d^{2} w}{d t^{2}}+\frac{e^{2 t}-v^{2}}{t^{4}} w=0\) is a variant of Bessel's differential equation (of the second kind) in its standard form \(x^{2}y'' + xy' + (x^{2} - v^{2})y = 0\). The variable change \(x = e^{1/t}\) can be utilized for transformation.

Variable Change

Perform the change of variable \(x = e^{1/t}\) and derive expressions for \(w', w''\) in terms of \(x\) and with respect to \(t\). Obtain expressions for \(w'\) (first derivative of \(w\)) and \(w''\) (second derivative of \(w\)) via the chain rule.

Substitution

Substitute the derived expressions for \(w'\) and \(w''\) into the original differential equation. The aim is to transform the original equation into the standard form of Bessel's differential equation.

Solving the Transformed Differential Equation

Once the equation is in the standard form of Bessel's differential equation, we recall that the solutions are in the form of Bessel's functions, \(J_{v}(x)\) and \(Y_{v}(x)\). The general solution is thus given by \(w=t\left[A J_{v}\left(e^{1 / t}\right)+B Y_{v}\left(e^{1 / t}\right)\right]\).

Key concepts

Bessel functions

Bessel functions are a family of solutions to Bessel's differential equation. This equation appears in many physical systems, especially in cylindrical and spherical symmetry. Bessel functions, often denoted as \( J_v(x) \) and \( Y_v(x) \), are named after the German mathematician Friedrich Bessel.
The function \( J_v(x) \) is known as the Bessel function of the first kind. It is of primary importance as these functions often emerge as solutions to problems with boundary conditions. The \( Y_v(x) \) functions are the Bessel functions of the second kind. They also solve Bessel's differential equation, complementing the concepts when singularities or different boundary conditions are involved.
These functions are crucial in mathematical physics because they provide the analytical description of various phenomena such as electromagnetic waves, heat conduction, and vibrations of circular membranes. Their oscillatory nature resembles sine and cosine functions, making them suitable for many periodic scenarios. Overall, Bessel functions are indispensable in solving and understanding complex differential equations that arise in physics.

differential equations

Differential equations are equations that relate a function to its derivatives. They play a vital role in modeling dynamic systems, such as populations, motion, and heat transfer. The core idea is to describe how a quantity changes over time or space.
In mathematical terms, a differential equation involves variables and their rates of change. The order of a differential equation is determined by the highest derivative present. For instance, in our exercise, we're dealing with a second-order differential equation because it involves the second derivative of \( w(t) \).
Solving a differential equation typically involves finding a function or functions that satisfy the equation. This process can be straightforward or complex, depending on the nature of the equation. A specific type that holds significant importance is the Bessel's differential equation, which features in numerous areas of mathematical physics, especially where cylindrical coordinates are involved.

mathematical physics

Mathematical physics is an interdisciplinary field that applies rigorous mathematical methods to solve problems in physics. It lies at the intersection of physical theories and mathematical frameworks to model real-world phenomena.
In mathematical physics, differential equations are commonly used to represent physical laws. For example, in electromagnetism, quantum mechanics, or fluid dynamics, these equations describe how physical systems evolve over time.
The Bessel equation in our problem illustrates the intersection of mathematics and physics. Bessel functions as solutions play a major role in characterizing waveforms and systems with symmetrical properties. This makes them highly applicable in disciplines such as acoustics, optics, and electromagnetic theory.
Through mathematical physics, complex systems can be analyzed, predicted, and understood, providing substantial insights into the natural world by harmonizing abstract mathematics with tangible physical interpretations.