If y = y(x), y ∈ [0, π/2] is the solution of the differential equation sec y (dy/dx) - sin(x+y) - sin(x-y) = 0, with y(0) = 0, then 5y'(π/2) is equal to ________.
If y = y(x), y ∈ [0, π/2] is the solution of the differential equation sec y (dy/dx) - sin(x+y) - sin(x-y) = 0, with y(0) = 0, then 5y'(π/2) is equal to ________.
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1 Answer
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sec y dy/dx = 2sinxcosy.
sec²y dy = 2sinx dx.
tan y = -2cosx + C.
y (0)=0 ⇒ 0=-2+C ⇒ C=2.
tan y = 2-2cosx.
y' = (-2sinx)/sec²y.
5y' (π/2) = 5 (2sin (π/2)/sec² (π/2)
sec²y dy/dx = 2sinx.
y' (π/2)? At x=π/2, tan y = 2. sec²y = 1+tan²y = 5.
5 (2sin (π/2) = 5 (2)=10.
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