Question
Let y=y(x) be the solution of the differential equation x ~d y ~d x -2 y=2+3 x, x (- 2 , 2 ), y(0)=- 7 4 . Then y ( 6 ) is equal to :
JEE Main
Mathematics
Differential Equations
2026
JEE Main 2026 (21 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. - 5 4
- B. - 5 2
- C. -3 3 -7
- D. -3 2 -7
Answer
B. - 5 2
Step-by-step solution
The differential equation is x dy dx - 2y = 2 + 3 x, or dy dx - 2 x y = (2 + 3 x) x. This is a linear ODE with integrating factor (x) = e^ -2 x . Multiplying both sides and integrating: d dx [ye^ -2 x ] = (2 + 3 x) x e^ -2 x Computing the integral: 2 x e^ -2 x dx = -e^ -2 x For the second term, using integration by parts: 3 x x e^ -2 x dx = 3 4 e^ -2 x (-2 x - 1) Thus: ye^ -2 x = -e^ -2 x + 3 4 e^ -2 x (-2 x - 1) + C Simplifying: y = - 7 4 - 3 2 x + Ce^ 2 x Using y(0) = - 7 4 gives C = 0. Therefore: y( 6 ) = - 7 4 - 3 2 1 2 = - 5 2
Related: Mathematics — Differential Equations · All PYQ Banks