Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
If the solution of the differential equation $\frac{dy}{dx} - y = 1 - e^{-x}$ and $y(0) = y_0$ has a finite value, when $x \to \infty$, then the value of $|2/y_0|$ is ______.
Step-by-Step Solution
Key Concept: Homogeneous differential equations can be solved by the substitution $y = vx$ to separate variables.
The perpendicular from origin to tangent at point $(x, y)$ has length $x$ where the tangent line satisfies $x \frac{dy}{dx} - y = x\sqrt{1 + (\frac{dy}{dx})^2}$. After simplification, this reduces to the homogeneous equation $x^2(\frac{dy}{dx})^2 - y^2 - 2xy\frac{dy}{dx} - x^2 + x^2(\frac{dy}{dx})^2 = \frac{x^2 - x^2}{2xy}\frac{dy}{dx}$. Substituting $y = vx$ transforms it to $v^2 + 1 = \frac{c}{x}$, giving $y^2 + x^2 = cx$. Using the initial condition that the curve passes through $(1, 1)$ yields $c = 2$, so the solution is $x^2 + y^2 - 2x = 0$.
Correct Answer: 4