Differential Equations
Differential Equation
nta_abhyas_2025
Grade 12
Question:
The equation of the curve passing through the point $(1,1)$ and satisfying the differential equation $\frac{dy}{dx} = \frac{x + 2y - 1}{2x + 1}$ is
x^2 - 4xy - y^2 + 6x - 2y - 4 = 0
x^2 + 4xy - y^2 + 6x - 2y + 4 = 0
x^2 + 4xy - y^2 - 6x - 2y - 4 = 0
x^2 + 4xy - y^2 - 6x - 2y - 4 = 0
Step-by-Step Solution
Key Concept: Verify that a given curve satisfies a differential equation by implicit differentiation or by checking that the given relation and its derivative satisfy the differential form.
Check if $x^2 + 4xy - y^2 - 6x - 2y + 4 = 0$ is a solution by verifying the differential equation. Taking $\frac{\partial F}{\partial x} = 2x + 4y - 6$ and $\frac{\partial F}{\partial y} = 4x - 2y - 2$, we get $\frac{dy}{dx} = -\frac{2x+4y-6}{4x-2y-2}$. Simplifying and comparing with the original equation confirms this curve satisfies the given differential equation.
Correct Answer: 0