Solving and Analyzing Differential Equations of the Form (2x^3y^2 - x^4y frac{dy}{dx} 0)

In the study of differential equations, it often becomes necessary to analyze and solve equations of various forms. One such form is the following:

[2x^3y^2 - x^4y frac{dy}{dx} 0]

Step-by-Step Solution

Let's proceed step-by-step to solve this differential equation.

Initial Equation

Starting with the given equation:

[2x^3y^2 - x^4y frac{dy}{dx} 0]

By rearranging the equation, we can simplify it:

[2x^3y^2 x^4y frac{dy}{dx}]

This can be further simplified as:

[2y - x frac{dy}{dx} 0]

Separation of Variables

To solve this, we can separate the variables (y) and (x) and integrate both sides:

[2 frac{dy}{y} -x frac{dx}{x}]

Integrating both sides, we get:

[2 ln|y| -ln|x^2| C]

This can be further simplified:

[2 ln|y| -2 ln|x| C]

Exponentiating both sides to solve for (y):

[y^2 e^{-2 ln|x| C}]

Using the property of exponents, we get:

[y^2 e^C cdot e^{-2 ln|x|}]

Since (e^C) is just a constant, we replace it with (A):

[y^2 A cdot |x|^{-2}]

This can be rewritten as:

[y^2 frac{A}{x^2}]

Therefore, the general solution is:

[y sqrt{frac{A}{x^2}} frac{sqrt{A}}{x}]

Alternative Approach

Another approach to solving the same differential equation is through direct separation:

[2x^3y^2 - x^4y frac{dy}{dx} 0]

Dividing through by (y cdot x^4), we get:

[ 2 frac{dx}{x} -frac{dy}{y}]

Integrating both sides:

[2 ln|x| -ln|y| C]

Which simplifies to:

[2 ln|x| ln|y| C]

Multiplying both sides by -1:

[-2 ln|x| - ln|y| -C]

This can be written as:

[ lnleft(frac{y}{x^2}right) C]

Exponentiating both sides:

[ y x^2 cdot e^C]

Since (e^C) is a constant, we can denote it as (A):

[y A x^2]

This is another general solution to the equation.

Special Case Solutions

It's also important to note the special solutions:

(x 0) (y 0)

These are immediate solutions that can be derived from the original equation.

Conclusion

Thus, the general solutions of the differential equation (2x^3y^2 - x^4y frac{dy}{dx} 0) are:

[y frac{sqrt{A}}{x}] [y A x^2]

where (A) is a constant. These solutions provide a comprehensive understanding of the behavior of the given differential equation.