Question

The equation of the curve passing through the origin and satisfying the equation
\[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2, \] is

• A. \( 3(1 + x^2) y = 4x^3 \)
• B. \( 3(1 - x^2) y = 4x^3 \)
• C. \( 3(1 + x^2) y = x^3 \)
• D. \( 4(1 - x^2) y = x^3 \)

Hint:

To solve first-order linear differential equations, try separating the variables and simplifying the equation step by step. Sometimes, recognizing standard forms like this one can help.

Answer 1

The Correct Option is A

Step 1: Understanding the given equation.
We are given the differential equation:
\[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2. \]
This is a first-order linear differential equation, and our goal is to find the equation of the curve that passes through the origin and satisfies this equation.

Step 2: Simplifying the equation.
We begin by isolating \( \frac{dy}{dx} \) from the given equation. Rearranging the terms:
\[ (1 + x^2) \frac{dy}{dx} = 4x^2 - 2xy. \]
Now, divide both sides by \( 1 + x^2 \) to solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{4x^2 - 2xy}{1 + x^2}. \]

Step 3: Using the method of separation of variables.
Next, we separate the variables \( x \) and \( y \). We can write:
\[ \frac{dy}{dx} = \frac{4x^2}{1 + x^2} - \frac{2xy}{1 + x^2}. \]
The term \( \frac{2xy}{1 + x^2} \) suggests that we will need to use the method of integrating factors or look for a suitable substitution to simplify further. Since this is a linear equation, we can attempt to solve it using known methods for linear first-order differential equations.

Step 4: Solving the equation.
At this point, solving the equation through integration gives the result:
\[ 3(1 + x^2) y = 4x^3. \]
This matches option (A).

Step 5: Conclusion.
Thus, the equation of the curve is:
\[ \boxed{3(1 + x^2) y = 4x^3}. \]