Re: [理工] [工數]-ODE
※ 引述《redwing119 (翼迷)》之銘言：
: 兩題變係數
: 1.(x+2)y"-(2x+5)y'+2y=(x+1)e^x
: 2.xy"+2y'-xy=2e^x
: 謝謝
1.(x+2)y"-(2x+5)y'+2y=(x+1)e^x
y=e^2xv=uv
y'=u'v+uv'
y''=u''v+2v'u'+uv''
(x+2)uv''+{2(x+2)u'-(2x+5)u}v'+{(x+2)u''-(2x+5)u'+2u)}v=(x+1)e^x
(x+2)v''+(2x+3)v'=(x+1)e^-x
e^2x (x+2)-1
{-----v'}'=--------e^x
x+2 (x+2)^2
e^2xv' e^x
-----=-----+c1
x+2 x+2
v'=e^-x+c1(x+2)e^-2x
v=-e^-x+c1'(2x+5)e^-2x+c2
y=c2e^2x+c1'(2x+5)-e^x
2.xy"+2y'-xy=2e^x
1
let y=---u
x
y'=-x^-2u+x^-1u'
y''=2x^-3u-2x^-2u'+x^-1u''
->u''-u=2e^x
->u=c1e^x+c2e^-2x+xe^x
->y=(c1e^x+c2e^-2x+xe^x)/x
--
3x10^6, 5.6x10^7, 4pix10^-7 ,8.85x10^-12
m d^m
P(x) =(1-x^2)^(m/2)--- ( P (x) )
l dx^m l
--
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