Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let f : (0, \infty) \to R be a function which is differentiable at all points of its domain and satisfies the condition 2 ′ x f (x) = 2xf (x) + 3 , with f (1) = 4. Then 2f (2) 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.
2 ′ x f (x) - 2xf (x) = 3 (1) 2 ′ x f (x) - 2xf (x) 3 ( ) = 2 2 2 2 (x ) (x ) d f (x) 3 \Rightarrow ( ) = 2 4 dx x x Integrating both sides f (x) 1 = - + C 2 3 x x 1 2 f (x) = - + Cx x put x = 1 4 = -1 + C \Rightarrow C = 5 1 2 f (x) = - + 5x x 1 2 Now 2 \times f (2) = 2 \times [- + 5 \times 2 ] 2 = 39
Correct Answer: 1