Re: [理工] [工數]-ODE
※ 引述《mdpming (+-*/)》之銘言：
: 1.
: 2
: y = 2xy' + y(y')
: 答案是
: y = +- ix 奇異解
: 2
: y = +- 根號 2c1*x + c1
令 y' = p
y = 2xp + yp^2
y(1 - p) (1 + p)
x = ───────────
2p
2p( p(1 - p^2) + y(-2pp') ) - 2p'y( 1 - p^2 )
1 = ──────────────────────────
4p^2
2 4 3 dp dp 3 dp 2
2 p - 2 p - 4 p y ── - 2 yp ── + 2yp ── - 4p
dy dy dy
────────────────────────────── = 0
4p^2
2 4 3 dp dp
- 2 p - 2 p - 2 y p ── - 2 y p ── = 0
dy dy
-2p^2 dy - 2p^4dy - 2yp^3dp - 2yp dp = 0
2 4 3
p dy + p dy + yp dp + yp dp = 0
3
p ( p dy + ydp ) + p ( ydp + pdy ) = 0
3
(p + p) d(yp) = 0
p = 0 (因為分母關係，所以不會成立)
2
p + 1 = 0 ， p = ±i
c
yp = c ， p = ──
y
代回去
2
y = 2xy' + y(y')
2
y = 2xp + yp
y = ±2xi - y
y = ±i x (Singular solution)
2 c x c 2
y = ─── + y (──)
y y
2 2
y = 2 c x + c (General solution)
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