If y = y(x) is the solution of the differential equation dy/dx + (tanx)y = sinx, 0 ≤ x ≤ π/3, with y(0) = 0, then y(π/4) equal to:
If y = y(x) is the solution of the differential equation dy/dx + (tanx)y = sinx, 0 ≤ x ≤ π/3, with y(0) = 0, then y(π/4) equal to:
Option 1 -
(1/2)logₑ2
Option 2 -
(1/(2√2))logₑ2
Option 3 -
(1/4)logₑ2
Option 4 -
logₑ2
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1 Answer
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Correct Option - 2
Detailed Solution:dy/dx + (tanx)y = sinx. This is a linear differential equation.
Integrating Factor (I.F.) = e^ (∫tanx dx) = e^ (ln|secx|) = secx.
The solution is y * I.F. = ∫ (sinx * I.F.) dx + C.
y * secx = ∫ (sinx * secx) dx = ∫tanx dx = ln|secx| + C.
Given y (0) = 0.
0 * sec (0) = ln|sec (0)| + C => 0 = ln (1) + C => C = 0.
So, y * secx = ln (secx).
y = cosx * ln (secx).
At x = π/4:
y = cos (π/4) * ln (sec (π/4) = (1/√2) * ln (√2) = (1/√2) * (1/2)ln (2) = ln (2) / (2√2).
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