Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
If $y = f(x)$ is solution of differential equation $\frac{dy}{x dx} - \frac{y}{x^2} = \frac{\sin^{-1}x}{2x^2}$, if $f(0) = 0$, then, $\frac{\pi}{f(1)}$ is equal to ____.
Step-by-Step Solution
Key Concept: Use integrating factor $x^2$ to convert the linear equation into an exact derivative form, then integrate using parts.
Given the differential equation $\frac{dy}{dx} = \frac{2y}{x} - \frac{\sin^{-1}x}{x}$, we find the integrating factor $e^{\int \frac{2}{x}dx} = e^{2\ln x} = x^2$. Multiplying through by $x^2$ gives $\frac{d}{dx}(yx^2) = \sin^{-1}x \cdot x^2$. Integration by parts and substitution yields $yx^2 = \frac{1}{2}(2x^2-1)\sin^{-1}x + \frac{1}{4}\sqrt{1-x^2} + c$. Using the initial condition $y(1) \cdot 1 = \frac{1}{4}(2-1)\cdot\frac{\pi}{2} + 0 = \frac{\pi}{8}$, we verify the solution.
Correct Answer: 8