微分方程(y∧2+x∧2)dy-xydx=0的通解是

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  • 设y/x=t,则y=xt,dy=xdt+tdx

    ∵(y²+x²)dy-xydx=0 ==>(y/x+x/y)dy-dx=0

    ==>(t+1/t)(xdt+tdx)=dx

    ==>x(t²+1)dt/t+(t²+1)dx=dx

    ==>x(t²+1)dt/t+t²dx=0

    ==>(1/t+1/t³)dt+dx/x=0

    ==>ln│t│-1/(2t²)+ln│x│=ln│C│ (C是积分常数)

    ==>xt=Ce^(1/(2t²))

    ==>y=Ce^(x²/(2y²)) (用t=y/x代换)

    ∴原微分方程的通解是y=Ce^(x²/(2y²)) (C是积分常数).