第六章常微分方程 解1:令y=p(x),y=p(x) y=(+y2)→p(x2=(+p( d +p)2+p2) x+c X+c 1+ p d +C1 dx=± x+c (x+ →(x+c1)+(y+ sdd 解2:令y=p(y),y=2p(y) t p =d→d 2(+p2 1+ (+C1 d(y+ →x+C,= 1-(+c (+c)2+(x+c2)2 第六章常微分方程第六章 常微分方程 第六章 常微分方程 解 1: 令 y = p(x) , y = p (x), ( )2 3 2 y = 1+ y ( ) ( ( ))2 3 2 p x = 1+ p x ( ) ( ) dx p d p dx p d p = + − = + − − 2 3 2 2 2 3 2 1 2 1 1 2 2 1 1 1 1 x c p dx p d = + + = + − − . ( ) ( ) ( ) 2 1 2 2 2 1 2 1 2 1 1 x c x c x c p p p − + + = + = + ( ) ( ) ( ) 2 1 2 1 2 1 2 1 2 1 dx x c x c d x c y c = − + − + + + = ( ) ( ) 1 2 2 2 x + c1 + y + c = 解 2: 令 y = p( y) , p(y) d y d p y = , ( )2 3 2 y = 1+ y ( )2 3 2 1 p dy dp p = + ( ) dy p dy d p d p = + − = + 2 2 3 2 2 1 1 2 1 1 2 1 1 y c p = − − + ( ) ( ) 2 1 2 2 1 1 y c y c p − + + = ( ) ( ) ( ) 2 1 2 1 2 1 2 1 2 1 dy y c y c d y c x c = − + − + + + = ( ) ( ) 1 2 2 2 y + c1 + x + c =