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Calculus
Differential Equations, Variable Separable
jee_main_2026_april_4_shift_2
Grade 12
Question:
The solution of the differential equation dy/dx = (x + y + 1)/(x + y - 1) that passes through (1, 0) satisfies y - x - ln|x + y| = k. Find k.
A. 0
B. 1
C. 2
D. -1
Step-by-Step Solution
Key Concept: Substitute v = x + y to reduce to variable separable form, then apply initial condition.
Step 1: Let v = x + y. Then dy/dx = dv/dx - 1. Step 2: dv/dx - 1 = (v + 1)/(v - 1) => dv/dx = (2v)/(v - 1). Step 3: (v - 1)/v dv = 2 dx => (1 - 1/v) dv = 2 dx. Step 4: v - ln|v| = 2x + C => x + y - ln|x + y| = 2x + C => y - x - ln|x + y| = C. Step 5: At (1, 0): 0 - 1 - ln|1| = C => C = -1.
Correct Answer:D
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