Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let x = x(y) be the solution of the differential equation y = (x - y dx ) sin( x y ), y > 0 and x(1) = \pi . Then dy 2 cos(x(2)) is equal to :
1 - 2(log 2) 2 e
1 - 2 (log 2) e
2 (log 2) - 1e
2(log 2) - 1 2 e
Step-by-Step Solution
Key Concept: Apply the core result for formation and solution of differential equations and simplify using the given constraints.
ydy = (xdy - ydx) sin( x ) (4) y dy xdy - ydx x = ( ) sin( ) 2 y y y dy x x = sin( )d (- ) y y y x \to ℓny = cos + C y \pi \pi x(1) = \Rightarrow 0 = cos + C \Rightarrow C = 0 2 2 x lny = cos y x but y = 2 \Rightarrow cos = ℓn2 2 2 x cos x = 2 cos - 1 2 2 = 2(ln 2) - 1
Correct Answer: 4