(x^2-4xy-2y^2)dx+(y^2-4xy-2x^2)dy=0

Jasonbradley

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Jun 17, 2024

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Solving the Differential Equation (x^2-4xy-2y^2)dx+(y^2-4xy-2x^2)dy=0

This article explores the solution to the given differential equation:

(x^2 - 4xy - 2y^2)dx + (y^2 - 4xy - 2x^2)dy = 0

This equation is a homogeneous differential equation, meaning it can be expressed in the form:

M(x, y)dx + N(x, y)dy = 0

where M and N are homogeneous functions of the same degree. In this case, both M and N are homogeneous functions of degree 2.

Solving the Homogeneous Equation

To solve this equation, we follow these steps:

1. Rewrite the equation in terms of a new variable: Let's introduce a new variable v = y/x. Then y = vx, and dy = vdx + xdv.

2. Substitute and simplify: Substitute these expressions for y and dy into the original equation:

   (x^2 - 4x(vx) - 2(vx)^2)dx + ((vx)^2 - 4x(vx) - 2x^2)(vdx + xdv) = 0

   Simplifying, we get:

   (x^2 - 4vx^2 - 2v^2x^2)dx + (v^2x^2 - 4vx^2 - 2x^2)(vdx + xdv) = 0

   Further simplification:

   x^2(1 - 4v - 2v^2)dx + x^2(v^2 - 4v - 2)(vdx + xdv) = 0

   Dividing both sides by x^2:

   (1 - 4v - 2v^2)dx + (v^2 - 4v - 2)(vdx + xdv) = 0

3. Separate variables: Rearrange the terms to get all the 'x' terms with 'dx' and all the 'v' terms with 'dv':

   (1 - 4v - 2v^2 + v^3 - 4v^2 - 2v)dx + (v^2 - 4v - 2)xdv = 0

   Simplifying:

   (1 - 6v - 6v^2 + v^3)dx + (v^2 - 4v - 2)xdv = 0

   Now, divide both sides by (1 - 6v - 6v^2 + v^3) and multiply both sides by (v^2 - 4v - 2):

   (v^2 - 4v - 2)dx / (1 - 6v - 6v^2 + v^3) = -dv

4. Integrate both sides: Integrate both sides with respect to their respective variables:

   ∫(v^2 - 4v - 2) / (1 - 6v - 6v^2 + v^3) dx = -∫dv

   The integral on the left side is a bit more complex and may require partial fraction decomposition. After integrating both sides, we obtain the solution in terms of x and v.

5. Substitute back for v: Finally, substitute v = y/x back into the solution to get the final solution in terms of x and y.

Solution of the Differential Equation

The solution to the differential equation (x^2-4xy-2y^2)dx+(y^2-4xy-2x^2)dy=0 is obtained by following the steps outlined above. While the integral on the left side of the equation can be solved using partial fractions, the final solution will involve complex expressions and is not easily presented in a concise form.

It's important to note that solving homogeneous differential equations often involves manipulating the equation through substitution and integration. This process can be complex, and the final solution might be expressed in a form that's not immediately intuitive. However, by understanding the steps involved, you can successfully solve these types of differential equations.