The system 2 x − y + z = 0 , x − 2 y + z = 0 , λ x − y + 2 z = 0 has infinitely many solutions. A continuous f satisfies f ( x ) + f ( x + 5 ) = 1 . Find 4 1 ∫ 0 2 λ f ( x ) d x .

Question

The system 2 x − y + z = 0 , x − 2 y + z = 0 , λ x − y + 2 z = 0 has infinitely many solutions. A continuous f satisfies f ( x ) + f ( x + 5 ) = 1 . Find 4 1 ​ ∫ 0 2 λ ​ f ( x ) d x .

TestHub · Accepted Answer

The determinant condition gives λ = 5 . Hence ∫ 0 10 ​ f = ∫ 0 5 ​ [ f ( x ) + f ( x + 5 )] d x = 5 . Dividing by 4 gives 1.25.

Mathematics - Differential Equation Question with Solution | TestHub

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