https://file.aiquickdraw.com/mathbot/file/172612215074101xl1xxx.jpg

The problem asks to find the general solution to the differential equation by treating **\( y \)** as the independent variable, rather than **\( x \)**. The given equation is:

\[
(1 + 2xy) \frac{dy}{dx} = 1 + y^2
\]

To solve this, we'll begin by switching the roles of **\( x \)** and **\( y \)**. This means we will express **\( x \)** as a function of **\( y \)**, and the derivative **\( \frac{dy}{dx} \)** becomes **\( \frac{dx}{dy} \)**.

### Step-by-step approach:

1. **Switch variables**: Rewrite the given equation by expressing **\( x \)** as a function of **\( y \)**.
   \[
   (1 + 2xy) \frac{dy}{dx} = 1 + y^2
   \]
   Rewriting in terms of **\( x \)** as the dependent variable:
   \[
   (1 + 2xy) \frac{dx}{dy} = \frac{1}{1 + y^2}
   \]

2. **Solve for \( x(y) \)**:
   This equation is separable, so we can solve it by integrating both sides.
   
   First, rewrite it as:
   \[
   \frac{dx}{dy} = \frac{1 + y^2}{1 + 2xy}
   \]
   We now proceed with solving this differential equation.

Let me proceed with the computations now.The general solution to the differential equation, treating \( y \) as the independent variable, is:

\[
x(y) = y + \frac{(4C_1^2 - 1)}{3}y^3 + \frac{4(12C_1^4 - 23C_1^2 + 2)}{15}y^5 + C_1 - C_1y^2 + 2C_1y^4(1 - C_1^2) + \dots
\]

This represents a series solution for \( x(y) \), where \( C_1 \) is the constant of integration.

Would you like further explanation on this solution or any of its components? Feel free to ask!

### Related Questions:
1. What is the meaning of treating \( y \) as the independent variable in a differential equation?
2. How do we solve separable differential equations step by step?
3. What is a series solution, and when is it used in differential equations?
4. How can we check if a solution is valid for a given differential equation?
5. What are some common techniques to solve first-order nonlinear differential equations?

### Tip:
Always check the domain of the solution to ensure it makes sense in the context of the problem (e.g., singularities or points where the function is not defined).

(1 + 2xy) dy/dx = 1 + y^2, solve by treating y as the independent variable.

Solve the differential equation (1 + 2xy) dy/dx = 1 + y^2 by treating y as the independent variable. This problem involves concepts from differential equations, including separation of variables and integration techniques, suitable for university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation