Differential Equations
Differential Equations
nta_pyq_2025_jan
Grade 12
Question:
Let a curve y = f (x) pass through the points (0, 5) and (log 2, k). If the curve satisfies the differential equation e 2(3 + y)e 2x dx - (7 + e 2x ) dy = 0 , then k 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.
2x dy 2(3 + y) ⋅ e = (3) dx 7 + e 2x 2x 2x dy 2ye 6 ⋅ e - = 2x 2x dx 7 + e 7 + e 2x 2e -\int dx 7+e2x I.F. = e 2x - ln(7+e ) \Rightarrow e 1 = 2x 7 + e 2x 1 6e y ⋅ = \int dx 2x 2 7 + e (7 + e 2x ) y -3 = 2x 2x 7 + e 7 + e + C \therefore y(0) = 5 5 -3 \Rightarrow = + C 8 8 \Rightarrow C = 1 2x \therefore y = -3 + 7 + e 2x y = e + 4 \therefore k = 8
Correct Answer: 3