Solving Differential Equations: The Approach to 2xydx (x^2 - 1)dy 0

Differential equations are a fundamental topic in mathematics, with applications spanning various fields such as physics, engineering, and economics. This article focuses on a specific type of differential equation, 2xydx (x^2 - 1)dy 0. We will explore various methods to solve this equation, starting with variable separation and moving through partial fractions, integration, and logarithmic forms.

Approach 1: Variable Separation

The given equation is:

2xydx (x^2 - 1)dy 0

To use the variable separation method, we can rearrange the equation:

2xydx (1 - x^2)dy

(2x/(1 - x^2))dx dy/y

Let's decompose the left-hand side using a substitution:

u x^2

du 2xdx

(du/(1 - u)) dy/y

Integrate both sides:

-ln|1 - u| ln|y| C

ln|1 - u| -ln|y| - C

ln|1 - u| ln|1/y| - C

1 - u Ce^(-ln|y|)

1 - u C/y

Substitute back u x^2:

1 - x^2 C/y

y C/(1 - x^2)

Approach 2: Integration

Another method is to directly integrate the given equation:

2xydx (x^2 - 1)dy 0

2xydx - (x^2 - 1)dy

(2x/(x^2 - 1))dx -dy/y

Integrate both sides:

∫(2x/(x^2 - 1))dx -∫(1/y) dy

ln|x^2 - 1| -ln|y| C

ln|x^2 - 1| ln|y| C

ln|y(x^2 - 1)| C

y(x^2 - 1) e^C

y Ce^{1/(x^2 - 1)}

Recall that Ce^{1/(x^2 - 1)} K (where C and K are arbitrary constants), so:

y K/(x^2 - 1)

Additional Solution Methods

Another way to solve the equation is using partial fractions and logarithmic forms:

2xydx (1 - x^2)dy

(2x/(1 - x^2))dx dy/y

Integrate both sides:

∫(2x/(1 - x^2))dx ln|y| - ln|c|

ln|1 - x^2| ln|y/c|

1 - x^2 cy

y (1 - x^2)/c

Let's consider another form:

y (1 - x^2)c

Conclusion

By using variable separation, integration, and partial fractions, we have derived the general solution to the differential equation 2xydx (x^2 - 1)dy 0. The solution is:

y K/(1 - x^2)

Where K is an arbitrary constant. This article provides a comprehensive overview of the methods used to solve such differential equations, which can be adapted to similar problems by following a similar process of variable separation, integration, and logarithmic forms.

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