Question

The solution of the differential equation \[ \frac{dy}{dx} + \frac{2yx}{1 + x^2} = \frac{1}{(1 + x^2)^2} \] is

• A. \( y(1 + x^2) = C + \tan^{-1} x \)
• B. \( \frac{y}{1 + x^2} = C + \tan^{-1} x \)
• C. \( y \log(1 + x^2) = C + \tan^{-1} x \)
• D. \( y(1 + x^2) = C + \sin^{-1} x \)

Hint:

When solving a first-order linear differential equation, identify the integrating factor and use it to simplify the equation before integrating.

Answer 1

The Correct Option is A

Step 1: Solve the differential equation.
The given differential equation is:
\[ \frac{dy}{dx} + \frac{2yx}{1 + x^2} = \frac{1}{(1 + x^2)^2}. \] This is a first-order linear differential equation. To solve it, we first identify the integrating factor, which is derived from the term \( \frac{2x}{1 + x^2} \).

Step 2: Integrate both sides.
We can integrate both sides after multiplying by the appropriate integrating factor. After solving, we find that the solution is of the form: \[ y(1 + x^2) = C + \tan^{-1} x. \]