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星星 asked in 科學數學 · 7 years ago

微分方程一題

y''-4xy'+4x^2y=xe^x^2

Update:

(D-2x+√2i)(D-2x-√2i)y是怎麼分的

Update 2:

怎麼知道要令yp=axe^(x^2)

為什麼不是yp=(ax+b)e^(x^2)

在幫我解qid=1614082804253

Update 3:

to 麻辣

你還沒有發覺你的yp令錯了嗎?

7 Answers

Rating
  • 7 years ago
  • 7 years ago

    原式 = [ ( D - 2x )^2 +2 ]y

    = [ ( D - 2x )^2 - ( √2i )^2 ]y

    = ( D - 2x + √2i )( D - 2x - √2i )y

  • 7 years ago

    to 麻辣:

    integrate tan x <> sec^2 x

    integrate tan x = - ln (cos x)

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  • 麻辣
    Lv 7
    7 years ago

    y"-4xy'+4x^2y=xe^x^2 (1) 齊次方程式y"-4xy'+4x^2y=0令y=e^t => y'=e^t*t'y"=e^t(t"+t'^2)代入上式裡面: 0=(t"+t'^2)-4xt'+4x^2

    令t'=u+2x => t"=u'+2代入上式裡面: 0=(u'+2)+(u^2+4xu+4x^2)-4x(u+2x)+4x^2=u'+u^2+2=du/(u^2+2)+dx=∫du/(u^2+2)+∫dx+c1=[atan(u/√2)]/√2+x-c1(c1-x)√2=atan(u/√2)tan[(c1-x)√2]=u/√2t'-2x=√2*tan[(c1-x)√2]t=2∫xdx+√2∫tan[(c1-x)√2]dx+ln(c2)=x^2-∫tan[(c1-x)√2]d[(c1-x)√2]+ln(c2)ln(y)=x^2-sec²[(c1-x)√2]+ln(c2)yh(x)=c2*exp{x^2-sec²[(c1-x)√2]}

    (2) 特殊解yp=a*x*e^(x^2)yp'=e^(x^2)*(a+2ax^2)yp"=e^(x^2)*(6ax+4ax^3)代入原式裡面,並把e^(x^2)刪除:x=(6ax+4ax^3)-4x(a+2ax^2)+4ax^2=2axa=1/2=> yp(x)=x*e^(x^2)/2

    (3) 一般解y(x)=yh+yp=c2*exp{x^2-sec²[(c1-x)√2]}+x*e^(x^2)/2......ans

    2014-08-28 07:41:15 補充:

    與版主答案有異

    但是過程似乎沒有錯誤

    有請高人指點

    2014-08-28 16:51:46 補充:

    感恩自由自在老師的指導

    tan積分修改:

    t=2∫xdx+√2∫tan[(c1-x)√2]dx+ln(c2)

    =x^2-∫tan[(c1-x)√2]d[(c1-x)√2]+ln(c2)

    ln(y)=x^2-ln{sec[(c1-x)√2]}+ln(c2)

    ln{ysec[(c1-x)√2]/c2}=x^2

    y*sec[(c1-x)√2]=exp(c^2*x^2)

    yh(x)=c3*cos[√2(c1-x)]*exp(x^2).......c3=exp(c^2)

    2014-08-28 16:52:12 補充:

    一般解

    y(x)=yh+yp

    =c3*cos[√2(c1-x)]*exp(x^2)+x*e^(x^2)/2

    =exp(x^2){c3*cos[√2(c1-x)]+x}......ans

    2014-08-28 16:56:13 補充:

    =exp(x^2){c3*cos(√2c1)*cos(√2x)+c3*sin(√2c1)*sin(√2x)}

    =exp(x^2){c4*cos(√2x)+c5*sin(√2x)}........ans

    =版主答案

    2014-08-28 16:58:23 補充:

    漏打x/2補充:

    y(x)=exp(x^2)*{c4*cos(√2x) + c5*sin(√2x) + x/2}........ans

    c4=c3*cos(√2c1)

    c5=c3*sin(√2c1)

  • 7 years ago

    沒有這麼複雜答案是y=e^x^2(c1cos√2x+c2sin√2x+x/2)

    2014-08-28 10:48:13 補充:

    你很厲害喔

    我剛剛還在找他哪裡有錯

    排版實在太難看了

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