Question

Consider the differential equation $$ z^{2}\left(z^{2}-1\right) y^{\prime \prime}+z\left(z^{2}-1\right) y^{\prime}+\frac{1}{16} y=0 $$ (a) Give the general solution, involving hypergeometric functions with argument $$ \frac{2 z}{z+1} $$ (b) Give the general solution, involving hypergeometric functions with argument $$ \frac{z-1}{2 z} $$

Short answer

For argument \(\frac{2z}{z+1}\): \( y(z) = C_1 {}_2F_1 ( \frac{1}{4}, \frac{1}{4}; \frac{1}{2}; \frac{2z}{z+1}) + C_2 (1-\frac{2z}{z+1})^{\frac{1}{2}} {}_2F_1 (\frac{1}{4},\frac{1}{4};\frac{3}{2};\frac{2z}{z+1})\). For argument \(\frac{z-1}{2z}\): \( y(z) = C_3 {}_2F_1 ( \frac{1}{4}, \frac{1}{4}; \frac{1}{2}; \frac{z-1}{2z}) + C_4 (1-\frac{z-1}{2z})^{\frac{1}{2}} {}_2F_1 ( \frac{1}{4}, \frac{1}{4}; \frac{3}{2}; \frac{z-1}{2z})\)

Step-by-step solution

- Identify standard form

Recognize that the given differential equation matches the form of a hypergeometric differential equation. The standard form is \( z(1 - z)y''(z) + (c - (a+b+1)z)y'(z) - aby(z) = 0 \).

- Rewrite the equation

Rewrite the given equation \( z^{2}(z^{2}-1) y'' + z(z^{2}-1) y' + \frac{1}{16} y = 0 \) in the form of a hypergeometric differential equation. Dividing by \(z^4 - z^2\) to isolate \(y''\), identify the parameters that match the standard form.

- Define the parameters

From the standard hypergeometric form, we identify: a = 1/4, b = 1/4 and c = 1/2. Verify these by substitution into the differential form.

- General solution using \(\frac{2z}{z+1}\)

Using the identified parameters and the change of variables, the general solution involving hypergeometric function with argument \( \frac{2z}{z+1} \) is given by \( y(z) = C_1 \, {}_2F_1 \left( \frac{1}{4}, \frac{1}{4}; \frac{1}{2}; \frac{2z}{z+1} \right) + C_2 \, (1-\frac{2z}{z+1})^{\frac{1}{2}} \, {}_2F_1 \left(\frac{1}{4}\,\frac{1}{4};\frac{3}{2};\frac{2z}{z+1}\right)\).

- General solution using \(\frac{z-1}{2z}\)

Using the same parameters and another change of variables, the general solution involving hypergeometric function with argument \(\frac{z-1}{2z}\) is given by: \( y(z) = C_3 \, {}_2F_1 \left( \frac{1}{4}, \frac{1}{4}; \frac{1}{2}; \frac{z-1}{2z} \right) + C_4 \, (1-\frac{z-1}{2z})^{\frac{1}{2}} \, {}_2F_1 \left( \frac{1}{4}, \frac{1}{4}; \frac{3}{2}; \frac{z-1}{2z} \right) \).

Key concepts

differential equations

Differential equations are mathematical equations that involve functions and their derivatives. They describe how a particular quantity changes over time or space.
In simpler terms, they are used to model situations where there is a relationship between a changing quantity and its rate of change. For example, in physics, they can represent how the velocity of an object changes over time under the influence of forces.
Depending on the problem, differential equations can be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs). In this exercise, we are dealing with an ODE as it involves a function of one variable and its derivatives.

general solution

The general solution of a differential equation provides a complete set of possible solutions to the equation.
For the given problem, once we identify that it's a hypergeometric differential equation, we can use hypergeometric functions to express the solution.
This specific general solution involves constants (like \(C_1, C_2\)) and functions (\({}_2F_1\)) that satisfy the original differential equation.
The form of these functions and constants might change based on the argument (like \(\frac{2z}{z+1}\) or \(\frac{z-1}{2z}\)).

hypergeometric functions

Hypergeometric functions are a class of special functions that play a crucial role in solving hypergeometric differential equations.
They are represented as \({}_2F_1(a, b; c; z)\) where \(a\), \(b\), and \(c\) are parameters that help define the function.
These functions are solutions to the hypergeometric differential equation of the form:
\[ z(1 - z)y''(z) + (c - (a+b+1)z)y'(z) - aby(z) = 0 \]
In our problem, the specific hypergeometric functions with parameters \(a = 1/4\), \(b = 1/4\), and \(c = 1/2\) help us express the general solution.

parameter identification

Identifying appropriate parameters \(a\), \(b\), and \(c\) is crucial to rewriting the given differential equation in its standard hypergeometric form.
This involves comparing the given equation with the standard form and adjusting the terms accordingly.
In our problem, through algebraic manipulation and comparison, we identified:

• \(a = 1/4\)
• \(b = 1/4\)
• \(c = 1/2\)

These parameters were verified by substitution into the rewritten equation.

change of variables

Changing variables is a powerful technique that simplifies complex differential equations into more manageable forms.
By choosing appropriate variable transformations, we can rewrite the original equation to match a known standard form.
In this exercise, different arguments like \(\frac{2z}{z+1}\) and \(\frac{z-1}{2z}\) were used to transform the original equation. These changes of variables allowed us to apply the hypergeometric function solutions to the problem.
Changing variables can make solving differential equations much easier and is a common technique in mathematical analysis.