Differential Equations
Homogeneous Differential Equations
IIT-JEE 1999
Grade 12
Question:
A curve passes through the point $(1, 1)$ and at every point on it the normal line passes through $(2, 0)$... Equivalently, it satisfies $x \frac{dy}{dx} - y = \sqrt{x^2 + y^2}$. Given $y(1) = 0$, the curve is:
(1) $y = \frac{x^2 + 1}{2}$
(2) $y = \frac{x^2 - 1}{2}$
(3) $y = \frac{(x - 1)^2}{2}$
(4) $y = x^2 - 1$
Step-by-Step Solution
Key Concept: Substitute $y = vx$ to convert the homogeneous equation into a variables-separable equation in $v$ and $x$, then integrate $\frac{dv}{\sqrt{1 + v^2}} = \frac{dx}{x}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)