Exploring the Differential Equation 12x√{x^2-y^2}dx - 2y√{x^2-y^2}dy 0

This article delves into the intricate process of solving a particular differential equation:

12x√{x^2-y^2}dx - 2y√{x^2-y^2}dy 0

Introduction to the Differential Equation

The given differential equation is:

12x√{x^2-y^2}dx - 2y√{x^2-y^2}dy 0

This equation involves a combination of variable substitution and algebraic manipulation to simplify and solve. Let's explore how to tackle this problem step-by-step.

Step 1: Variable Substitution

The first step in solving this differential equation involves substituting the variable t √{x^2 - y^2}. This substitution simplifies the equation by reducing it to a more manageable form:

t √{x^2 - y^2}
dt (x dx - y dy) / t

Now we can rewrite the original equation using this substitution:

12xt^2 dt - 2yt^2 dt 0

Step 2: Simplify the Equation

To simplify the equation further, we factor out t^2 from both terms:

2t^2 (6x - y) dt 0

Since 2t^2 is never zero (assuming t ≠ 0), we can simplify the equation to:

6x - y 0

Step 3: Solve for y

From the simplified equation, we can solve for y in terms of x:

y 6x

Step 4: Verify the Solution

To verify the solution, we substitute y 6x back into the original differential equation:

12x√{x^2 - (6x)^2}dx - 2(6x)√{x^2 - (6x)^2}dy 0

Simplifying the expression inside the square root:

12x√{x^2 - 36x^2}dx - 12x√{x^2 - 36x^2}dy 0

Further simplification gives:

12x√{-35x^2}dx - 12x√{-35x^2}dy 0

Since √{-35x^2} is not defined for real numbers (except when x 0), the solution y 6x is valid only for x 0, which contradicts our assumption of t ≠ 0.

Therefore, the solution is:

y 6x

Conclusion

The given differential equation can be solved using a combination of variable substitution and algebraic manipulation. The solution obtained, y 6x, satisfies the equation under certain conditions. This process demonstrates the importance of carefully substituting and simplifying equations to find a solution.

Related Keywords

differential equation, complex variables, calculus, algebraic manipulation