Question

Find the Wronskian of a given set \(\left\\{y_{1}, y_{2}\right\\}\) of solutions of $$x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\nu^{2}\right) y=0$$ given that \(W(1)=1\). (This is Bessel's equation.)

Short answer

Answer: The equation relating the values of the Wronskian, functions, and their derivatives at \(x = 1\) is: $$1 = \frac{1}{2}\left(y_1(1)y_2(1) -2\nu^2 y_1(1)y_2'(1) - y_1'(1)y_2(1) + 2\nu^2 y_1'(1)y_2(1)\right)$$

Step-by-step solution

Differentiate both solutions

Since we don't have explicit expressions for \(y_1\) and \(y_2\), we will first differentiate both sides of the given Bessel's equation with respect to x, using the solutions \(y_1\) and \(y_2\). Differentiating Bessel's equation with respect to x, we get: $$2x y^{\prime \prime} + x^{2} y^{\prime\prime\prime} + y^{\prime} + x y^{\prime\prime} - 2\nu^{2} y^{\prime} = 0$$ Rearranging, we obtain an expression for the second derivative: $$y^{\prime\prime} = -\frac{2\nu^2 y'(x) - y(x)}{x^2 + x}$$

Find the Wronskian

Now, using the expression for the second derivative from step 1, we can find the Wronskian \(W(y_1,y_2)\) as defined: $$W(y_1,y_2) = y_1 y_2' - y_1' y_2$$ Replace \(y_1''\) and \(y_2''\) with the expressions we found in step 1: $$W(y_1,y_2) = y_1\left(-\frac{2\nu^2 y_2'(x) - y_2(x)}{x^2 + x}\right) - y_1'\left(-\frac{2\nu^2 y_1'(x) - y_1(x)}{x^2 + x}\right)$$ Simplifying, we get: $$W(y_1,y_2) = \frac{1}{x^2+x}\left(y_1y_2(x) -2\nu^2 y_1(x)y_2'(x) - y_1'(x)y_2(x) + 2\nu^2 y_1'(x)y_2(x)\right)$$

Use the given condition

Now we will use the given condition that \(W(1) = 1\). Let's substitute \(x = 1\) in the expression for the Wronskian: $$1 = \frac{1}{2}\left(y_1(1)y_2(1) -2\nu^2 y_1(1)y_2'(1) - y_1'(1)y_2(1) + 2\nu^2 y_1'(1)y_2(1)\right)$$ Without explicit expressions for \(y_1\) and \(y_2\), we cannot simplify further. However, we now have an equation that relates the values of the functions and their derivatives at \(x = 1\). This equation can be used to find numerical values of the Wronskian under specific conditions for given solutions of the Bessel's equation.

Key concepts

Bessel's Equation

Bessel's equation is a fundamental concept in mathematics and physics, primarily arising in problems exhibiting cylindrical symmetry. It is a type of second-order linear differential equation expressed as:\[ x^2 y^{\prime\prime} + x y^{\prime} + (x^2 - u^2) y = 0 \] Here, \( u \) is a parameter, often referred to as the order of the Bessel function. The solutions to this equation, known as Bessel functions, are critical in fields like engineering, physics, and applied mathematics. They occur in contexts such as wave propagation, heat conduction, and electrical signal processing.Bessel's equation can be challenging to solve directly, but its solutions are well-studied and documented. These solutions, typically denoted as \( J_u(x) \) and \( Y_u(x) \), represent two linearly independent solutions of the Bessel equation of order \( u \). These are used to form general solutions for specific boundary conditions in various applications.

Differential Equations

Differential equations are equations involving functions and their derivatives. They play a crucial role in expressing physical phenomena in mathematical form. A simple differential equation might look like:\[ y' = f(x, y) \]There are several key points to understand about differential equations:

• First and Second Order: The order of a differential equation is determined by the highest derivative in the equation. Bessel's equation is a second-order differential equation because it involves the second derivative \( y^{\prime\prime} \).
• Linear vs Non-linear: Linear differential equations have solutions that can be written as linear combinations of functions, while non-linear equations can model more complex systems but are typically harder to solve.
• Applications: Differential equations model diverse natural phenomena such as motion, growth, shrinkage, and spreading of disease.

The study and solution of these equations are foundational for students in fields like engineering, physics, and mathematics as they provide insight and predictive power for real-world problems.

Solutions of Differential Equations

Understanding solutions of differential equations, such as those seen in Bessel's equation, is vital for students solving complex problems involving multiple variables.The solutions can be:

• General Solutions: These incorporate constants and account for all potential solution curves without specific initial conditions.
• Particular Solutions: These satisfy the differential equation along with specific initial or boundary conditions. It is practical for deriving solutions applicable to real-world situations, like determining wave patterns or electrical currents.
• Exact Solutions: These closed-form solutions provide precise results and insights but may not always be feasible for complex or non-linear differential equations.
• Numerical Solutions: When analytical solutions elude us, numerical methods such as Euler's method, the Runge-Kutta methods, and others provide approximate results useful in computing solutions for complex systems.

For Bessel's equation, the solutions often involve Bessel functions \( J_u(x) \) and \( Y_u(x) \) since they meet both the equation and any applicable boundary conditions.