Question

Solution of differential equation \[ x^2-1+\left(\frac{x}{y}\right)^{-1}\frac{dy}{dx} +\frac{x^2}{2!}\left(\frac{dy}{dx}\right)^2 +\frac{x^3}{3!}\left(\frac{dy}{dx}\right)^3+\cdots=0 \] is

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

Hint:

Identify known series expansions to simplify differential equations.

Answer 1

The Correct Option is A

Step 1: Series represents expansion of \(e^{(dy/dx)}\).
Step 2: Simplifying differential equation gives: \[ \frac{d(y^2)}{dx}=2x\ln x \]
Step 3: Integrating: \[ y^2=x^2(\ln x-1)+C \]