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

In this article, we will discuss how to solve a specific type of differential equation, namely (x^2 - 4xy - 2y^2 dx y^2 - 4xy - 2x^2 dy 0). This is an exact differential equation, and we will walk through the steps to simplify and solve it.

Identifying Exact Differential Equations

An exact differential equation is of the form (M(x, y) dx N(x, y) dy 0) where (M) and (N) are functions of (x) and (y) and the equation is exact if (frac{?M}{?y} frac{?N}{?x}).

Solving the Given Equation

The given equation is:

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

To solve this, we start by rewriting the equation in the standard form:

(int M dx int N dy C)

Where (M x^2 - 4xy - 2y^2) and (N y^2 - 4xy - 2x^2).

Integrating the Terms

We begin with the integration of (M dx):

(int x^2 - 4xy - 2y^2 dx frac{x^3}{3} - 2x^2y - 2xy^2 int N dy)

Now, we integrate the terms that do not contain (x) in the (N) expression:

(int y^2 dy frac{y^3}{3})

Thus, combining both integrals, we get:

(frac{x^3}{3} - 2x^2y - 2xy^2 frac{y^3}{3} C)

(boxed{frac{x^3}{3} - 2x^2y - 2xy^2 frac{y^3}{3} C})

Using the Substitution Method

Alternatively, we can use the substitution (y vx), where (v frac{y}{x}) and (dy v dx x dv). Substituting (y) and (dy) in the original equation:

(x^2 - 4vx^2 - 2v^2x^2 dx (vx)^2 - 4x(vx) - 2x^2 (v dx x dv) 0)

Simplifying this, we get:

(1 - 4v - 2v^2 dx v^2 - 4v - 2 dx vx dv 0)

Further simplification gives:

(1 - 6v - 6v^2 dx (v^3 - 4v^2 - 2v) dv 0)

Which can be rearranged as:

(frac{dx}{x} frac{-v^3 - 4v^2 - 2v dv}{v^3 - 6v - 6v^2})

And applying partial fractions:

(frac{-v^3 - 4v^2 - 2v dv}{v^3 - 6v - 6v^2} frac{A}{v 1} frac{B(v^2 - 7v - 1)}{v^2 - 7v - 1})

After solving, we find (A -frac{1}{3}), (B -frac{1}{3}) and (C 0).

The integration leads us to:

(ln x - frac{ln (v^3 - 7v 1) - ln (v^2 - 7v 1)}{3} ln C)

Finally, subsituting back (v frac{y}{x}), we get:

(x^3 cdot left(frac{y}{x}right) cdot left(frac{y^2}{x^2} - frac{7y}{x} 1right) C)

(boxed{xy cdot (x^2 - 7xy - y^2) C})

Conclusion

In conclusion, we have successfully solved the exact differential equation (x^2 - 4xy - 2y^2 dx y^2 - 4xy - 2x^2 dy 0) using both direct integration and substitution methods. Understanding and mastering exact differential equations is crucial in solving a wide range of problems in mathematics and physics.