y#39;#39;-2yy#39;3(三次方)=0y#39;(0)=-1y(0)=1解初值(可降价的高阶微分方程)
<p>问题：y#39;#39;-2yy#39;3(三次方)=0y#39;(0)=-1y(0)=1解初值(可降价的高阶微分方程)<p>答案：↓↓↓<p class="nav-title mt10" style="border-top:1px solid #ccc;padding-top: 10px;">罗付华的回答：<div class="content-b">网友采纳 ∵令y'=p,则y"=pdp/dy 代入原方程,得pdp/dy-2yp^3=0 ==>p(dp/dy-2yp^2)=0 ∴p=0,或dp/dy-2yp^2=0 ∵p=0不满足初始条件,舍去 ∴dp/dy-2yp^2=0 ==>dp/p^2=2ydy ==>-1/p=y^2-C1(C1是常数) ==>-1/y'=y^2-C1 ==>-dx/dy=y^2-C1 ==>dx=-y^2+C1 ==>x=C1y-y^3/3+C2(C2是常数) ∵y(0)=1,y'(0)=-1 ∴代入x=C1y-y^3/3+C2,得C1=0,C2=1/3 故原方程满足初始条件的特解是x=(1-y^3)/3.
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