Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let y = f (x) be the solution of the differential equation dy xy x +4x dx + 2 x -1 = , -1 < x < 1 such that f (0) = 0. \sqrt1-x2 1/2 If 6 \int -1/2 f (x)dx = 2\pi - \alpha then \alpha is equal to _______ . 2
Step-by-Step Solution
Key Concept: Apply the core result for formation and solution of differential equations and simplify using the given constraints.
I.F. e - 2 \int 1-x2 dx = e - 1 2 ln(1-x ) 2 = \sqrt1 - x 2 (27) y \times \sqrt1 - x = \int (x + 4x) dx = 2 6 x 7 + 2x 2 + c 7 Given y(0) = 0 \Rightarrow c = 0 7 x 2 +2x 7 y = \sqrt1-x2 1 x7 2 1 +2x 2 Now, 6 \int 2 1 7 dx = 6 \int 2 1 2x dx - \sqrt1-x2 - \sqrt1-x2 2 2 1 2 2 x = 24 \int dx 0 \sqrt1 - x 2 Put x = sin \theta dx = cos \thetad\theta \pi 2 6 sin \theta = 24 \int cos \thetad\theta 0 cos \theta \pi \pi 6 1 - cos 2\theta sin 2\theta 6 = 24 \int ( ) d\theta = 12[\theta - ] 2 2 0 0 \pi \sqrt3 = 12 ( - ) 6 4 = 2\pi - 3\sqrt3 2 2 \alpha = (3\sqrt3) = 27
Correct Answer: 27