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Most Important Selected Qs for JEE AdvancedMathematicsDifferential Equations

Let f(1)+g(1)=9 e ; f(x)=-x^2 g^ (x) ; g(x)=-x^2 f^ (x) . If ₁^4 f^ (x)+g(x) x^2 d x=k (e-e^ 1 4 ) , then find the value of k .

Correct answer

9

Step-by-step solution

aligned & f(1)+g(1)=9 e, f(x)=-x^2 g^ (x) and g(x)=-x^2 f^ (x) & I= ₁^4 f(x)+g(x) x^2 d x= ₁^4 -x^2 (f^ (x)+g^ (x) ) x^2 d x= ₁^4- (f^ (x)+g^ (x) ) d x & =-[f(x)+g(x)]₁^4=-(f(4)+g(4)-9 e) aligned Now, f(x)+g(x)=-x^2 (g^ (x)+f^ (x) ) aligned & g^ (x)+f^ (x) f(x)+g(x) = -1 x^2 f(x)+g(x)=a e^ 1 x f^ (1)+g(1)=9 e a=9 & f(x)+g(x)=9 e^ 1 x & I=- (9 e^ 1 4 -9 e )=9 (e-e^ 1 4 )=k (e-e^ 1 4 ) k=9 aligned

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