大一微积分(一阶线性微分方程)(y²-6x)y'+2y=0,求它的通解?
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![大一微积分(一阶线性微分方程)(y²-6x)y'+2y=0,求它的通解?](/uploads/image/z/13415728-40-8.jpg?t=%E5%A4%A7%E4%B8%80%E5%BE%AE%E7%A7%AF%E5%88%86%EF%BC%88%E4%B8%80%E9%98%B6%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%EF%BC%89%EF%BC%88y%26%23178%3B-6x%EF%BC%89y%27%2B2y%EF%BC%9D0%2C%E6%B1%82%E5%AE%83%E7%9A%84%E9%80%9A%E8%A7%A3%3F)
大一微积分(一阶线性微分方程)(y²-6x)y'+2y=0,求它的通解?
大一微积分(一阶线性微分方程)
(y²-6x)y'+2y=0,求它的通解?
大一微积分(一阶线性微分方程)(y²-6x)y'+2y=0,求它的通解?
∵(y^2-6x)y'+2y=0 ==>(y^2-6x)y'=-2y
==>(y^2-6x)dy/dx=-2y
==>dx/dy=(y^2-6x)/(-2y)
==>dx/dy=3x/y-y/2
==>dx/dy-3x/y=-y/2
∴先解齐次方程dx/dy-3x/y=0的通解
∵dx/dy-3x/y=0 ==>dx/dy=3x/y
==>dx/x=3dy/y
==>ln|x|=3ln|y|+ln|C| (C是积分常数)
==>x=Cy³
∴齐次方程dx/dy-3x/y=0的通解是x=Cy³ (C是积分常数)
于是,应用“常数变易法”,设原微分方程的通解为x=uy³ (u是关于y的函数)
∵dx/dy=y³du/dy+3uy²
∴把它代入dx/dy-3x/y=-y/2
得y³du/dy+3uy²-3uy³/y=-y/2
==>y³du/dy+3uy²-3uy²=-y/2
==>y³du/dy=-y/2
==>y²du/dy=-1/2
==>du=-dy/(2y²)
==>u=1/(2y)+C (C是积分常数)
把u=1/(2y)+C代入x=uy³,得x=[1/(2y)+C]y³=y²/2+Cy³
故原微分方程的通解是x=y²/2+Cy³ (C是积分常数).
转自heanmen