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JEE Main202131 Aug 2021Morning ShiftMathematicsDefinite IntegrationActual

If x ϕ ( x ) = ∫ 5 x 3 t 2 - 2 ϕ ' ( t ) d t , x > - 2 , ϕ ( 0 ) = 4 , then ϕ ( 2 ) is

Correct answer

4

Step-by-step solution

Given that x ϕ ( x ) = ∫ 5 x 3 t 2 - 2 ϕ ' ( t ) d t ϕ ( 0 ) = 4 ,    x > - 2 Differentiating on both the sides and using Newton Leibnitz rule ⇒ x ϕ ' ( x ) + 1 ϕ ( x ) = 3 x 2 - 2 ϕ ' ( x ) ∵ d d x U V   =   U d V d x   +   V d U d x   ;   d d x ∫ f x g x h x d x   =   g ' x h g x - f ' x h f x ⇒ ( x + 2 ) ϕ ' ( x ) + ϕ ( x ) = 3 x 2 It is a linear differential equatio

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