Solving the Equation x2y2 1/x2 1/y2 4 in Calculus: Techniques and Applications

Calculus is a powerful tool for solving complex equations, and one such equation that can be tackled through various methods is x2y2 1/x2 1/y2 4. This article will guide you through the steps to solve this equation using algebraic manipulation and the Cauchy-Schwarz inequality, providing a clear understanding of the process and its applications.

Introduction to the Equation

The equation at hand, x2y2 1/x2 1/y2 4, involves variables and their reciprocals. To simplify the problem, it is often helpful to introduce new variables. Let's start by setting a x2 and b y2.

Simplifying the Equation

By substituting a and b, the original equation transforms into:

a b 1/a 1/b 4

We can rewrite the terms involving reciprocal values:

a b (a b)/ab 4

This can be rearranged as:

(a b) (a b)/ab 4

Let s a b and p ab. Then we have:

s s/p 4

Multiplying through by p, we get:

sp s 4p

Rearranging gives:

sp - 4p s 0

This can be factored as:

s(p - 1) 4p

Therefore, we find:

s 4p / (p - 1)

Applying the Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a powerful tool that states:

(a b)(1/a 1/b) ≥ 1 12 4

This implies that:

(x2 1/x2) (y2 1/y2) ≥ 4

Equality holds when a b. Let's set a b. Then our equation simplifies to:

2a 2?1/a 4

This simplifies to:

a 1/a 2

Multiplying through by a, we get:

a2 - 2a 1 0

This can be factored as:

(a - 1)2 0

Thus, we find:

a 1

Since a x2 and b y2, we have:

x2 1 and y2 1

Taking the square root, we find:

x 1 or x -1

y 1 or y -1

Thus, the solutions for x and y are:

(1, 1), (1, -1), (-1, 1), (-1, -1)

Conclusion

In summary, the pairs (x, y) that satisfy the equation x2y2 1/x2 1/y2 4 are:

(1, 1), (1, -1), (-1, 1), (-1, -1)

Understanding these techniques can apply to a broader range of equations and is a fundamental skill in calculus and mathematical problem-solving.

References:

[1] Algebraic Methods and Inequalities in Calculus, Chapter 3, Section 4