Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let f be a differentiable function such that 2(x + 2) f (x) - 3(x + 2) = 10 \int (t + 2)f (t)dt, x \ge 0. Then f (2) 2 2 x 0 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.
x 2 2 2(x + 2) f (x) - 3(x + 2) = 10 \int (t + 2)f (t)dt 0 (19) Differentiating both side 2 4(x + 2)f (x) + 2f (x)(x + 2) - 6(x + 2) = 10(x + 2)f (x) dy = (x + 2) - 3y = 3 dx 1 dy dx \int = \int 3 y + 1 x + 2 ln |y + 1 = 3 ln |x + 2 ∣ + ln c 3 y + 1 = (x + 2) c 3 ∵ y(0) = 2 5 \Rightarrow = c 16 5 3 \therefore y = (x + 2) - 1 16 5 y(2) = \times 64 - 1 = 19 16
Correct Answer: 19