The following differential equation represents a family of what curves?

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

To determine the family of curves represented by the differential equation 

\[ \frac{dx}{y} = \frac{dy}{1-2x}, \]

we start by rewriting the equation in a more manageable form. Cross-multiplying gives:

\[ y \, dy = (1 - 2x) \, dx. \]

Now we integrate both sides separately:

1. Integrate the left side with respect to \( y \):

\[ \int y \, dy = \frac{y^2}{2} + C_1. \]

2. Integrate the right side with respect to \( x \):

\[ \int (1 - 2x) \, dx = x - x^2 + C_2. \]

Since both sides are indefinite integrals, we can combine the constants \( C_1 \) and \( C_2 \) into a single constant \( C \):

\[ \frac{y^2}{2} = x - x^2 + C. \]

To simplify, multiply both sides by 2:

\[ y^2 = 2x - 2x^2 + C'. \]

where \( C' = 2C \) is a new constant. 

Rearrange the terms to put it in a standard form of a conic section:

\[ y^2 + 2x^2 - 2x = C'. \]

This represents a family of curves. Specifically, this is the general form of a conic section, which could be an ellipse, parabola, or hyperbola, depending on the value of \( C' \). The specific type of conic is determined by the discriminant \( B^2 - 4AC \) of the quadratic equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). 

In this case, since the equation has no \( xy \) term (i.e., \( B = 0 \)) and both \( A \) and \( C \) are positive (i.e., \( A = 2 \) and \( C = 1 \)), this suggests the curves are ellipses or parabolas, depending on \( C' \).

To confirm:

- If \( C' < 1 \), the equation represents an ellipse.
- If \( C' = 1 \), it represents a parabola.
- If \( C' > 1 \), it represents a hyperbola.

Would you like more details on how to analyze the type of conic sections further or have any questions?

### Related Questions

1. What are the steps to solve a differential equation of the form \( M(x, y)dx + N(x, y)dy = 0 \)?
2. How do you identify the type of conic section from its general equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)?
3. What is the significance of the discriminant \( B^2 - 4AC \) in determining the nature of conic sections?
4. How can you convert a differential equation to its standard form?
5. What methods can be used to solve a first-order differential equation?
6. What are some real-world applications of conic sections in physics and engineering?
7. How can we interpret the geometric meaning of a differential equation?
8. What are the conditions for a differential equation to represent a particular type of conic section?

### Tip

When dealing with differential equations, always check if they can be simplified or transformed into a more familiar form to make integration and solution finding easier.

Explore the differential equation dx/y = dy/(1-2x) and understand its representation of a family of conic sections. Learn how to integrate and analyze the equation step-by-step, identifying whether the curves formed are ellipses, parabolas, or hyperbolas based on the discriminant. Suitable for advanced high school students and those interested in calculus and algebra.

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