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 Question:
This is a linear differential equation of the form \( \frac{dy}{dx} + \text{P}y = \text{Q} \). We solve it using an integrating factor (I.F.).
Step 2: Key Formula or Approach:
Standard form: \( \frac{dy}{dx} + \frac{2x}{1+x^2}y = \frac{4x^2}{1+x^2} \).
I.F. \( = \text{e}^{\int \text{P}dx} \).
Step 3: Detailed Explanation:
1. \( \text{P} = \frac{2x}{1+x^2} \).
I.F. \( = \text{e}^{\int \frac{2x}{1+x^2}dx} = \text{e}^{\ln(1+x^2)} = 1 + x^2 \).
2. General solution:
\( y \times (\text{I.F.}) = \int \text{Q} \times (\text{I.F.}) dx + \text{C} \)
\( y(1 + x^2) = \int \frac{4x^2}{1+x^2}(1 + x^2) dx + \text{C} \)
\( y(1 + x^2) = \int 4x^2 dx + \text{C} = \frac{4x^3}{3} + \text{C} \).
3. Since curve passes through origin \( (0, 0) \):
\( 0(1+0) = 0 + \text{C} \Rightarrow \text{C} = 0 \).
4. Equation: \( y(1 + x^2) = \frac{4x^3}{3} \Rightarrow 3(1 + x^2)y = 4x^3 \).
Step 4: Final Answer:
The equation is \( 3(1 + x^2)y = 4x^3 \).