Question

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

• A. $y=log_{e}(x)+c$
• B. $y=(log_{e}x)^{2}+c$
• C. $y=\pm\sqrt{(log_{e}x)^{2}+2c}$
• D. $xy=x^{y}+k$

Hint:

Solution of the differential equation $x=1+xydydx+\frac(xy)/2!(dy/dx)+(xy)/3!(dy/dx)+...$ is

Answer 1

The Correct Option is C

Step 1: Concept
Recognize the RHS as the Taylor series expansion of $e^{xy\frac{dy}{dx}}$.
Step 2: Analysis
The equation is $x = e^{xy\frac{dy}{dx}}$, which simplifies to $\log x = xy \frac{dy}{dx}$.
Step 3: Evaluation
Rearranging gives $y \, dy = \frac{\log x}{x} \, dx$. Integrating both sides: $\int y \, dy = \int \log x \, d(\log x)$.
Step 4: Conclusion
$\frac{y^2}{2} = \frac{(\log x)^2}{2} + c \Rightarrow y^2 = (\log_e x)^2 + 2c$, thus $y = \pm\sqrt{(log_e x)^2 + 2c}$.
Final Answer: (c)