Question

The following two \(R(r)\) equations arise in various separation of variables problems in polar, cylindrical, or spherical coordinates: $$\begin{aligned}&r \frac{d}{d r}\left(r \frac{d R}{d r}\right)=n^{2} R,\\\&\frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=l(l+1) R.\end{aligned}$$ There are various ways of solving them: They are a standard kind of equation (often called Euler or Cauchy equations see Chapter \(8,\) Section \(7 \mathrm{d}\) ); you could use power series methods; given the fact that the solutions are just powers of \(r,\) it is easy to find the powers. Choose any method you like, and solve the two equations for future reference. Consider the case \(n=0\) separately. Is this necessary for \(l=0 ?\)

Short answer

For the first equation, the solution is \[ R(r) = C_1 r^n + C_2 r^{-n} \]. For the second equation, the solution is \[ R(r) = D_1 r^l + D_2 r^{-l-1} \]. Separate consideration for \( l = 0 \) is not necessary.

Step-by-step solution

Understanding the First Equation

The first equation is \[ r \frac{d}{d r}\left(r \frac{d R}{d r}\right)=n^{2} R \] This is an Euler-Cauchy differential equation.

Substituting Solution Form

Assume a solution of the form \[ R(r) = r^k \]. Then, compute the first and second derivatives: \[ \frac{dR}{dr} = kr^{k-1} \] and \[ \frac{d^2R}{dr^2} = k(k-1)r^{k-2} \].

Applying Derivatives in the First Equation

After substituting and simplifying, the equation is \[ r \frac{d}{d r}(r \cdot kr^{k-1}) = n^2 r^k \]. Simplify it to get \[ r \cdot k(k-1)r^{k-2} + r \cdot kr^{k-1} = n^2 r^k \]. Further simplification yields \[ k(k-1)r^k + k r^k = n^2 r^k \].

Finding the Exponent

Combine like terms: \[ k^2 r^k = n^2 r^k \]. Thus, \[ k^2 = n^2 \], which gives \[ k = \pm n \]. So, the general solution for the first equation is \[ R(r) = C_1 r^n + C_2 r^{-n} \].

Understanding the Second Equation

The second equation is \[ \frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=l(l+1) R \] This, too, is an Euler-Cauchy differential equation.

Substituting Solution Form in Second Equation

Assume a solution of the form \[ R(r) = r^k \]. Then, compute the first derivative: \[ \frac{dR}{dr} = kr^{k-1} \] and the second derivative: \[ \frac{d^2 R}{dr^2} = k(k-1)r^{k-2} \].

Applying Derivatives in the Second Equation

Substitute into the second equation: \[ \frac{d}{d r}\left(r^2 \cdot kr^{k-1}\right) = l(l+1)r^k \]. Simplify to \[ r^2 \cdot k(k-1) r^{k-2} + 2r \cdot kr^{k-1} = l(l+1) r^k \], or \[ k(k+1) r^k = l(l+1) r^k \].

Finding the Exponent in the Second Equation

Combine like terms to obtain \[ k(k+1) = l(l+1) \]. Thus, \[ k = l \text{ or } k = -l-1 \]. So, the general solution for the second equation is \[ R(r) = D_1 r^l + D_2 r^{-l-1} \].

Considering the Case n=0

For the first equation, if \( n = 0 \), the solution simplifies to \[ R(r) = C_1 + C_2 \frac{1}{r} \]. For \( l = 0 \) in the second equation, the solution simplifies to \[ R(r) = D_1 + D_2 \frac{1}{r} \]. Since the structure is the same, separate consideration for \( l = 0 \) is not necessary.

Key concepts

Separation of Variables

Separation of Variables is a fundamental technique used to solve differential equations. It involves rewriting a problem so that each variable occurs on a different side of the equation. The solution process usually includes:

• Identifying the variables that need to be separated.
• Rewriting the equation in a separable form.
• Integrating both sides of the equation with respect to their respective variables.

This technique is particularly useful in solving problems related to heat transfer, wave equations, and potential equations in physics and engineering.

Polar Coordinates

Polar coordinates represent a point in the plane using a distance from a reference point and an angle from a reference direction. Instead of using \(x\) and \(y\) coordinates, polar coordinates use \(r\) (the radius) and \(θ\) (the angle). This system is convenient for problems involving circular or rotational symmetry.

• In polar coordinates, a point \(P\) is denoted as \( (r, θ) \).
• Conversion from Cartesian to polar coordinates: \( r = \sqrt{x^2+y^2} \), \(θ = \arctan\left(\frac{y}{x}\right) \)

Polar coordinates simplify the solving process of the Euler-Cauchy differential equation in circular domains.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional extension of polar coordinates. They are useful for solving problems with symmetry around a central axis. A point is represented by \( (r, \theta, z) \), where:

• \( r \) is the radial distance from the origin to the projection of the point onto the \(xy\)-plane.
• \( \theta \) is the angular coordinate.
• \( z \) is the height above the \(xy\)-plane.

Equations in cylindrical coordinates can simplify the solving process of differential equations, particularly those with cylindrical symmetry.

Spherical Coordinates

Spherical coordinates are ideal for solving problems where the symmetry is about a point. They represent points using three parameters: \( (ρ, θ, φ) \). Here:

• \( ρ \) is the distance from the origin to the point.
• \( \theta \) is the azimuthal angle in the \(xy\)-plane, from the positive \(x\)-axis.
• \( φ \) is the polar angle from the positive z-axis.

Spherical coordinates are often used in solving problems in electrostatics, gravitation, and quantum mechanics as they simplify the mathematical form of the governing equations.

Power Series Methods

Power series methods involve expressing solutions to differential equations as an infinite sum of terms. Each term is a power of the independent variable. This method is powerful for solving linear differential equations with variable coefficients.

• Given a differential equation, assume a solution of the form \( y(x) = \sum_{n=0}^\infty a_n x^n \).
• Substitute this series into the differential equation.
• Match coefficients of like powers of \( x \) to determine the series coefficients \( a_n \).

Power series solutions are particularly useful when the exact analytic solution is complex or when working near a point such as zero.

Differential Equations

Differential equations involve functions and their derivatives. They are classified based on order and linearity:

• The order of a differential equation is determined by the highest derivative present.
• Linear differential equations can be written as Linear Combination of Derivatives = Function.

Understanding differential equations is essential because they describe various physical systems, such as motion, growth, heat, and waves.

Exponents

Exponents are used to denote repeated multiplication of a number by itself. They work as a shorthand and are crucial in solving differential equations. Some key points include:

• \( a^n = a \times a \times ... \times a \) (n times)
• They help express solutions to Euler-Cauchy equations by allowing the assumption of solutions of the form \( R(r) = r^k \).

Understanding exponents simplifies the integration, differentiation, and solving of various algebraic and differential equations.