JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

The given differential equation is: (x^2 - x x^2 - 1 )dy + (y(x - x^2 - 1 ) - x)dx = 0 Dividing the entire equation by dx and rearranging, we get: x(x - x^2 - 1 ) dy dx + y(x - x^2 - 1 ) = x Dividing by x(x - x^2 - 1 ), we obtain a linear differential equation: dy dx + 1 x y = 1 x - x^2 - 1 The integrating factor (I.F.) is: I.F. = e^ 1 x dx = e^ x = x Multiplying the differential equation by the integrating factor x, we get: x dy dx + y = x x - x^2 - 1 d dx (xy) = x(x + x^2 - 1 ) (x - x^2 - 1 )(x + x^2 - 1 ) d dx (xy) = x^2 + x x^2 - 1 x^2 - (x^2 - 1) d dx (xy) = x^2 + x x^2 - 1 Integrating both sides with respect to x: d(xy) = (x^2 + x x^2 - 1 ) dx xy = x^3 3 + (x^2 - 1)^ 3/2 3 + C Given y(1) = 1, we substitute x = 1 and y = 1: 1(1) = 1^3 3 + 0 + C C = 1 - 1 3 = 2 3 So, the particular solution is: xy = x^3 3 + (x^2 - 1)^ 3/2 3 + 2 3 To find y( 5 ), we substitute x = 5 : 5 y( 5 ) = ( 5 )^3 3 + (5 - 1)^ 3/2 3 + 2 3 5 y( 5 ) = 5 5 3 + 4^ 3/2 3 + 2 3 5 y( 5 ) = 5 5 3 + 8 3 + 2 3 = 5 5 + 10 3 y( 5 ) = 5 3 + 10 3 5 = 5 + 2 5 3 Since 5 2.236, we have 2 5 4.472. y( 5 ) 5 + 4.472 3 = 9.472 3 3.157 The greatest integer less than y( 5 ) is 3. Answer: 3