Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let y = y(x) be the solution of the differential equation (xy - 5x \sqrt1 + x ) dx + (1 + x ) dy = 0, y(0) = 0. 2 2 2 Then y(\sqrt3) is equal to
\sqrt 15 2 5\sqrt3
2
2\sqrt2
\sqrt 14 3
Step-by-Step Solution
Key Concept: Apply the core result for formation and solution of differential equations and simplify using the given constraints.
(1 + x ) 2 dy + xy = 5x \sqrt1 + x 1 2 (2) dx 2 dy xy 5x + = 2 dx 1 + x \sqrt1 + x 2 2 x ln(1+x ) \int dx 2 \therefore I.F. = e 1+x2 = e 2 = \sqrt1 + x 2 5x 2 2 \therefore y \sqrt1 + x = \int ⋅ \sqrt1 + x dx \sqrt1 + x2 2 5x 2 2 \therefore y \sqrt1 + x = \int ⋅ \sqrt1 + x dx \sqrt1 + x2 3 5x 2 y \sqrt1 + x = + C 3 ∵ y(0) = 0 \Rightarrow 0 = 0 + C \Rightarrow C = 0 3 5x \therefore y = 2 3\sqrt 1 + x 15\sqrt3 5\sqrt 3 y(\sqrt3) = = 32 2
Correct Answer: 2