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MAT244--2019F => MAT244--Test & Quizzes => Quiz-1 => Topic started by: Changhao Jiang on September 27, 2019, 03:05:02 PM

Title: TUT0102 Quiz1
Post by: Changhao Jiang on September 27, 2019, 03:05:02 PM
Question: show it is homogeneous and solve it dy/dx = (x+3y)/(x-y)
Solution´╝Ü dy/dx = (1+3y/x)/(1-y/x) in this form, which shows it is homogeneous
               let u = y/x, y = ux
               differentiate on both sides with x, we get
               dy/dx = u + xdu/dx
               from above, dy/dx = (1+3u)/(1-u)
               then (1+3u)/(1-u) =  u + xdu/dx
               xdu/dx = (1+3u)/(1-u) - u = (u+1)^2/(1-u)
               (1/x) dx = ((1-u) / (u+1)^2) du
               integrating on both sides, we get
               ln|x| = - 2/u+1 - ln|u+1| + C
               -ln|x| - 2/(y/x+1) - ln(y/x+1) + C =0
               - ln(y+x) - 2x/y+x + C = 0