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JEE Main202226 Jul 2022Evening ShiftMathematicsDifferential EquationsActual

Suppose y = y x be the solution curve to the differential equation d y d x - y = 2 - e - x such that lim x → ∞ y x is finite. If a and b are respectively the x - and y - intercept of the tangent to the curve at x = 0 , then the value of a - 4 b is equal to _______.

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Given, d y d x - y = 2 - e - x Linear differential equation, So, I . F .   = e - ∫ dx = e - x Now solution of differential equation is given by y × I F = ∫ 2 - e - x × I F d x ⇒ y e - x = ∫ 2 - e - x e - x d x ⇒ y e - x = ∫ 2 e - x - e - 2 x d x ⇒ y e - x = - 2 e - x + e - 2 x 2 + C ⇒ y = - 2 + e - x 2 + C e x Given lim x → ∞ y is finite So, lim x → ∞ - 2 + e - x 2 + C · e x → finite This is possible only when C = 0 He

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