Solving a Differential Equation

  1. 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!
     
  2. jcsd
  3. dextercioby

    dextercioby 12,292
    Science Advisor
    Homework Helper

    Looks ok to me.
     
  4. Thanks! I just remembered that I can check these in my calculator as well.... >.<
     
  5. derivative y=√x+√x
    y'=?
     
  6. 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...?
     
  7. under sqrtx is also +sqrtx
     
  8. Char. Limit

    Char. Limit 1,986
    Gold Member

    Don't hijack other problems with your own.
     
  9. sorry
     
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