微分方程y'=y^2/(xy-x^2)和y'=xye^(x^2)Iny

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微分方程y''=y^2/(xy-x^2)和y''=xye^(x^2)Iny微分方程y''=y^2/(xy-x^2)和y''=xye^(x^2)Iny微分方程y''=y^2/(xy-x^2)和y''=xye^(x^

微分方程y'=y^2/(xy-x^2)和y'=xye^(x^2)Iny
微分方程y'=y^2/(xy-x^2)和y'=xye^(x^2)Iny

微分方程y'=y^2/(xy-x^2)和y'=xye^(x^2)Iny
1.微分方程y'=y^2/(xy-x^2)
令y/x=t,则y'=xt'+t
代入原方程,得
y'=(y/x)²/((y/x)-1)
==>xt'+t=t²/(t-1)
==>xt'=t/(t-1)
==>dx/x=(1-1/t)dt
==>ln│x│=t-ln│t│+ln│C│ (C是积分常数)
==>xt=Ce^t
==>x(y/x)=Ce^(y/x)
==>y=Ce^(y/x)
故原方程的通解是y=Ce^(y/x) (C是积分常数).
2.微分方程y'=xye^(x^2)Iny
∵y'=xye^(x²)Iny
==>dy/(ylny)=xe^(x²)dx
==>d(lny)/lny=[e^(x²)/2]d(x²)
==>ln│lny│=e^(x²)/2+ln│C│ (C是积分常数)
==>lny=Ce^[e^(x²)/2]
∴原方程的通解是lny=Ce^[e^(x²)/2] (C是积分常数).

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