Question

  
Find the general solution or particular solution of each the following DE's 1) (y-y2 tanx)dx + (2y+tanx)dy=0 2) (x2+y2+x)dx + xydy-0 i y(-1)-1 4) For the initial value problem y' + xy - xy? ex2 ; y(0)-1 Find the explicit solution if y>0 dy dae dy
  

Answer 1

(2y^3 - x^3) dx + 3xy^2 dy = 0, y(1) = 1 = > (2y^3 - x^3)/3xy^2 = -dy/dx = > dy/dx = x^3/3xy^2 - 2y^3/3xy^2 = x^2/3y^2 - 2y/3x = > dy/dx + 2/3x middot y = x^2 y^-2/3 = > y^2 dy/dx + 2/3x y^3 = x^2/3 = > 3y^2 dy/dx + 2/x y^3 = x^2 put y^3 = v = > 3y^2 dy = dv = > dv/dx + 2/x v = x^2 factor = e^integral 2/3 dx = e^2lnx = x^2 this solution is v, x^2 = integral x^2 times x^2 dx = x^5/5 + c = > v = x^3/5 + cx^-2 = > y^3 = x^3/5 + cx^-2 y(1) = 1 = > 1 = 1/5 + c = > c = 4/5 = > y^3 = x^3/5 + 4/5x^-2 (y^4 - 2xy) dx + 3x^2 dy = 0, y(2) = 1 = > dy/dx = 2xy/3x^2 - y^4/3x^2 = > dy/dx + 2y/3x = -y^4/3x^2 = > 3y^-4 dy/dx + 2y^-3/x = -1/x^2, put y^-3 = t 3y^-4 dy = dt = > -dt/dx + 2t/x =

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