Let y = y(x) be the solution of the differential equation xdy = (y + x³cosx)dx with y(π) = 0, then y(π/2) is equal to:
Let y = y(x) be the solution of the differential equation xdy = (y + x³cosx)dx with y(π) = 0, then y(π/2) is equal to:
Option 1 -
π²/4 - π/2
Option 2 -
π²/4 + π/2
Option 3 -
π²/2 - π/4
Option 4 -
π²/2 + π/4
Asked by Shiksha User
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1 Answer
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Correct Option - 2
Detailed Solution:xdy - ydx = x³cosxdx
(xdy-ydx)/x² = xcosxdx
d (y/x) = xcosxdx
y/x = ∫xcosxdx = xsinx - ∫sinxdx = xsinx + cosx + c
y = x²sinx + xcosx + cx
y (π) = 0 + π (-1) + cπ = 0 ⇒ c = 1
y = x²sinx + xcosx + x
y (π/2) = (π/2)² (1) + 0 + π/2 = π²/4 + π/2
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