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## Homework Statement

Solve:

2 * √(x) * (dy/dx) = cos^2(y)

y(4) = π/4

## Homework Equations

TRIGONOMETRIC RECIPROCAL IDENTITY: sec(u) = 1 / cos(u)

arctan( 1 ) = π / 4

## The Attempt at a Solution

This is a separable differential equation.

2 * √(x) * (dy/dx) = cos^2(y)

2 * √(x) * dy = cos^2(y) * dx

[2 / cos^2(y)] * dy = [1 / √(x)] * dx

TRIGONOMETRIC RECIPROCAL IDENTITY: sec(u) = 1 / cos(u)

[2 * sec^2(y)] * dy = x^(-1/2) * dx

∫ [2 * sec^2(y)] * dy = ∫ x^(-1/2) * dx

2 * tan(y) = 2 * √(x) + C

y(x) = arctan( √(x) + C) <<< General Solution

NOTE: arctan( 1 ) = π / 4

y(4) = arctan( √(4) + C )

y(4) = arctan( 2 + C)

C = -1

y(x) = arctan( √(x) - 1 ) <<< Particular Solution

Is that all correct? Thank you!