Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade None
Question:
Solution of the differential equation $x = 1 + xy\frac{dy}{dx} + \frac{x^2y^2}{2!}\left(\frac{dy}{dx}\right)^2 + \frac{x^3y^3}{3!}\left(\frac{dy}{dx}\right)^3 + ......$ is:
y = ln(x) + c
y = (ln x)^2 + c
y = x√((ln x)^2 + c)
xy = x^y + c
Step-by-Step Solution
Key Concept: Separating variables after logarithmic transformation allows direct integration of both sides.
The equation $x = e^{xy(dy/dx)}$ reduces to $\log x = xy\frac{dy}{dx}$. Integrating both sides: $\int y dy = \int \frac{\log_e x}{x}dx$. This gives $\frac{y^2}{2} = \frac{(\log_e x)^2}{2} + C$, so $y = \pm\sqrt{(\log_e x)^2 + C}$.
Correct Answer: 3