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.