dx尀dy=2x-y*2 微分方程求通解~高数！

这是一阶线性微分方程，由通解公式：
x=e^(2y)(C-∫y^2e^(-2y)dy)

=e^(2y)(C+(1/2)∫y^2de^(-2y)
=e^(2y)(C+(1/2)(y^2e^(-2y)-∫ye^(-2y)dy)
=e^(2y)(C+(1/2)(y^2e^(-2y)-(1/2)∫yde^(-2y))
=e^(2y)(C+(1/2)(y^2e^(-2y)-(1/2)ye^(-2y)+(1/2)∫e^(-2y)dy)
=e^(2y)(C+(1/2)(y^2e^(-2y)-(1/2)ye^(-2y)-(1/4)e^(-2y))
=Ce^(2y)+(1/2)(y^2-y-2)

如果是 dx\dy=2x-2y
u=2x-2y 2y=2x-u 2dy/dx=2-du/dx
所以
2dy/dx=2-du/dx=4x-4y=2u
2-du/dx=2u
du/dx=-2(u-1)
-2x+C=ln|u-1|
u-1=2x-2y-1=Ce^-2x
2x-2y-1=Ce^-2x

如果是 dx\dy=2x-y^2
先解齐次方程
dx/dy=2x;
则x=Ce^2y
dx/dy=dC/dy *e^2y+2Ce^2y =dC/dy *e^2y+2x =2x-y^2
dC/dy =-y^2*e^-2y
C=∫-y^2*e^-2y dy 用分部积分法
=1/2* y^2*e^-2y-∫y*e^-2ydy
=1/2* y^2*e^-2y+1/2y*e^-2y-∫e^-2ydy
=1/2* y^2*e^-2y+1/2*y*e^-2y+1/2*e^-2y +C1
代入通解
2x=y^2+y+1+C1*e^2y