1. The problem statement, all variables and given/known data Solve: 2 * √(x) * (dy/dx) = cos^2(y) y(4) = π/4 2. Relevant equations TRIGONOMETRIC RECIPROCAL IDENTITY: sec(u) = 1 / cos(u) arctan( 1 ) = π / 4 3. 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!
Um... what? Well, if y=√(x)+√(x) then y=2√(x) and y'=4x^(3/2)/3 but you need to separate the variables first, then integrate not derive, so I don't see how that's relevant...?