Question

Find the Wronskian of two solutions of the given differential equation without solving the equation. \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\alpha(\alpha+1) y=0, \quad\) Legendre's equationn

Short answer

#Answer# The Wronskian of two solutions of Legendre's equation is a constant, and it can be determined by the initial conditions of the problem. That is, \(W(u,v) = C\), where \(C\) is a constant.

Step-by-step solution

Differentiate the Wronskian with respect to \(x\)

The Wronskian \(W(u,v)\) is defined by \(W(u,v) = u(x) v'(x) - u'(x) v(x)\). First, differentiate both sides with respect to \(x\): $$W'(u,v) = u(x) v''(x) + u'(x) v'(x) - u''(x) v(x) - u'(x) v'(x)$$

Substitute Legendre's equation in place of \(u''\) and \(v''\)

Now, we substitute the expressions of \(u''\) and \(v''\) from Legendre's equation into the expression for \(W'(u,v)\). First of all, we rewrite Legendre's equation in terms of \(y''\): $$y'' = \frac{1}{(1-x^2)}\left[2xy' - \alpha(\alpha+1)y\right]$$ Applying the above expression for \(u\) and \(v\): $$u''=\frac{1}{(1-x^2)}\left[2xu'-\alpha(\alpha+1)u\right],\qquad v''=\frac{1}{(1-x^2)}\left[2xv'-\alpha(\alpha+1)v\right]$$ Now substitute these expressions into the expression for \(W'(u,v)\): $$W'(u,v) = u(x) \left(\frac{1}{(1-x^2)}\left[2xv'-\alpha(\alpha+1)v\right]\right) + u'(x) v'(x) - v(x) \left(\frac{1}{(1-x^2)}\left[2xu'-\alpha(\alpha+1)u\right]\right) - u'(x) v'(x)$$

Simplify the expression for \(W'(u,v)\)

Now we simplify the expression for \(W'(u,v)\): $$W'(u,v) =u'(x)v'(x) -u'(x)v'(x) + \frac{u(x) v(x)}{(1-x^2)}\alpha(\alpha+1) - \frac{u(x) v(x)}{(1-x^2)}\alpha(\alpha+1)$$ We observe that all terms cancel out except the last ones. So, $$W'(u,v) = 0$$

Integrate to find the Wronskian

Since \(W'(u,v) = 0\), the Wronskian must be a constant. To find this constant, we integrate both sides with respect to \(x\): $$\int W'(u,v) \, dx = \int 0 \, dx$$ Which gives us, $$W(u,v) = C$$ where \(C\) is a constant. So, the Wronskian of two solutions of Legendre's equation is a constant, and it can be determined by the initial conditions of the problem.

Key concepts

Legendre's Equation

Legendre's equation is a second-order linear differential equation often encountered in physics and engineering, particularly in solving problems related to spherical harmonics and potential theory. The general form of Legendre's equation is expressed as:
\[(1-x^2)y'' - 2xy' + \alpha(\alpha+1)y = 0\]Here, \(y\) represents the function we are solving for, \(y'\) its first derivative, and \(y''\) its second derivative. The constant \(\alpha\) relates to the order of the Legendre polynomial, a special function defined by this equation. Legendre's equation is self-adjoint, lending itself well to various solution methods.

• Appears in physics, especially in quantum mechanics and electromagnetism.
• Generates solutions known as Legendre polynomials.

Understanding Legendre's equation is crucial for students as it builds foundational knowledge for more complex topics.

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. They are widely used to model real-world phenomena in sciences and engineering. Differential equations can be classified into several types:

• Ordinary Differential Equations (ODEs): Involve functions of a single variable and their derivatives.
• Partial Differential Equations (PDEs): Involve multiple independent variables.

``` The focus in this exercise is an ODE: Legendre's equation. Such equations describe the rate at which one quantity changes with respect to another and can express laws of physics and changes in dynamic systems. A proper grasp of differential equations allows for understanding various natural and engineered systems' behaviors.
Being linear, Legendre's equation fits a broad category where methods like the Wronskian can aid in verifying independence in solutions.

Solution Methods

Finding solutions to differential equations can vary greatly between simple and complex equations. For linear ODEs like Legendre's equation, several methods exist:

• Direct Integration: Useful for simple equations where the derivative can be directly integrated.
• Series Solutions: Involved in simplifying complex equations, especially those leading to polynomials.
• Wronskian Method: Helps determine linear independence of solutions without explicitly finding them.

In this exercise, the Wronskian method is utilized. Starting by expressing the derivatives within the context of Legendre's equation, simplification reveals that the Wronskian is constant for the solutions, indicating linear independence. Solution methods for differential equations provide structured approaches to exploring possible behaviors and confirming solution characteristics.

Initial Conditions

Initial conditions are crucial in uniquely determining solutions of differential equations. They provide specific values at certain points, ensuring that among all possible solutions, the right one is selected.
For Legendre's equation and many others, solutions aren't fully characterized without initial conditions. Initial conditions might specify:

• The value of the function at a starting point.
• The value of its derivative at that point.

In the context of the exercise, the integration of the Wronskian with respect to \(x\) resulted in a constant. This constant is pivotal because initial conditions dictate its value, cementing the specific form of the solution. Initial conditions thus convert general solutions into the specific solution needed for a real-world application.