Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let y = y(x) be the solution of the differential equation 2 cos x dy = sin 2x - 4y sin x, x \in (0, \pi ) . If dx 2 y( \pi 3 ) = 0 , then y ( ′ \pi 4 ) + y( \pi 4 ) is equal to ________ ________.
Step-by-Step Solution
Key Concept: Apply the core result for formation and solution of differential equations and simplify using the given constraints.
dy + 2y tan x = sin x (1) dx 2 \int tan xdx 2 I.F. = e = sec x sin x 2 y sec x = \int dx 2 cos x = \int tan x sec xdx = sec x + C C = -2 2 y = cos x - 2 cos x \pi 1 y( ) = - 1 4 \sqrt2 ′ y = - sin x + 4 cos x sin x \pi 1 ′ y ( ) = - + 2 4 \sqrt2 \pi \pi ′ y ( ) + y( ) = 1 4 4
Correct Answer: 1