急 有关微分方程的几道题1．
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急 有关微分方程的几道题1．
急 有关微分方程的几道题
1．
急 有关微分方程的几道题1．
设p=y'=>p'=y''
p'+2xp^2=0
分离变量
-dp/p^2=2xdx
1/p=x^2+C
p=1/(x^2+C)
y'|x=0=-1/2=>-1/2=1/C=>C=-2
y'=1/(x^2-2)
y=(ln|x+根号2|-ln|x-根号2|)/2根号2+C
y|x=0=1=>C=1
y=(ln|x+根号2|-ln|x-根号2|)/2根号2+1
∵y''+2xy'²=0 ==>-dy'/dx=2xy'²
==>-dy'/y'²=2xdx
==>d(1/y')=d(x²)
==>1/y'=x²+C1 (C1是积分常数)
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∵y''+2xy'²=0 ==>-dy'/dx=2xy'²
==>-dy'/y'²=2xdx
==>d(1/y')=d(x²)
==>1/y'=x²+C1 (C1是积分常数)
==>y'=1/(x²+C1)
又当x=0时，y'=-1/2.则1/C1=-1/2 ==>C1=-2
∴y'=1/(x²-2)
∴y=∫dx/(x²-2)
=√2/4∫[1/(x-√2)-1/(x+√2)]dx
=√2/4[ln|x-√2|-ln|x+√2|]+C2 (C2是积分常数)
=√2/4ln|(x-√2)/(x+√2)|+C2
∵当x=0时，y=1.则C2=1
∴y=√2/4ln|(x-√2)/(x+√2)|+1
故原微分方程的特解是：y=√2/4ln|(x-√2)/(x+√2)|+1
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