solve differential equation (x+2y²) dx/dy = y what the solution of equation

To solve the differential equation \((x + 2y^2) \frac{dx}{dy} = y\), we can start by rearranging it into a more manageable form. This equation is a first-order ordinary differential equation, and we can separate the variables to find a solution.

Rearranging the Equation

First, let's rewrite the equation:

\((x + 2y^2) \frac{dx}{dy} = y\)

We can express \(\frac{dx}{dy}\) as:

\(\frac{dx}{dy} = \frac{y}{x + 2y^2}\)

Separating Variables

Next, we can separate the variables \(x\) and \(y\). This means we want all terms involving \(x\) on one side and all terms involving \(y\) on the other side:

\(dx = \frac{y}{x + 2y^2} dy\)

Integrating Both Sides

Now, we can integrate both sides. However, integrating the right side directly can be tricky. We can rewrite the equation as:

\(\int (x + 2y^2) dx = \int y dy\)

Let's integrate the left side with respect to \(x\) and the right side with respect to \(y\).

Left Side Integration

The left side becomes:

\(\int (x + 2y^2) dx = \frac{x^2}{2} + 2y^2 x + C_1\)

Right Side Integration

The right side becomes:

\(\int y dy = \frac{y^2}{2} + C_2\)

Combining the Results

Now we can set the two integrals equal to each other:

\(\frac{x^2}{2} + 2y^2 x = \frac{y^2}{2} + C\)

where \(C = C_2 - C_1\) is a constant of integration.

Final Form of the Solution

To express this in a more standard form, we can rearrange it as:

\(x^2 + 4y^2 x - y^2 = 2C\)

This is the implicit solution to the differential equation. Depending on the context, you might want to solve for \(x\) explicitly, but this form is often sufficient for many applications.

Summary

• The original equation was rearranged and separated into integrable parts.
• Both sides were integrated, resulting in an implicit solution.
• The final form can be expressed as \(x^2 + 4y^2 x - y^2 = 2C\).

Understanding how to manipulate and solve differential equations like this one is crucial in many fields, including physics and engineering, where such equations often model real-world phenomena. If you have any further questions or need clarification on any steps, feel free to ask!