求下列微分方程的通解,xdy/dx=(yIn^2)y,[（y+1)^2]dy/dx+x^3=0,dy/dx=2^(x+y),6x+y求下列微分方程的通解，xdy/dx=(yIn^2)y，[（y+1)^2]dy/dx+x^3=0，dy/dx=2^(x+y)，我有急用

Question

求下列微分方程的通解,xdy/dx=(yIn^2)y,[（y+1)^2]dy/dx+x^3=0,dy/dx=2^(x+y),6x+y
求下列微分方程的通解，xdy/dx=(yIn^2)y，[（y+1)^2]dy/dx+x^3=0，dy/dx=2^(x+y)，我有急用

Answer 1

1.求xdy/dx=yIn²y通解
∵xdy/dx=yIn²y ==>dy/(yIn²y)=dx/x
==>d(lny)/In²y=dx/x
==>-1/lny=ln│x│+C (C是积分常数)
经检验y=1也是原方程的解
∴原方程的通解是y=1或-1/lny=ln│x│+C (C是积分常数)；
2.求[(y+1)²]dy/dx+x³=0通解
∵[(y+1)²]dy/dx+x³=0 ==>[(y+1)²]dy=-x³dx
==>(y+1)³/3=C/3-x^4/4 (C是积分常数)
==>(y+1)³=C-3x^4/4
∴原方程的通解是(y+1)³=C-3x^4/4 (C是积分常数)；
3.求dy/dx=2^(x+y)通解
∵dy/dx=2^(x+y) ==>dy/dx=(2^x)(2^y)
==>dy/2^y=2^xdx
==>e^(-yln2)dy=e^(xln2)dx
==>e^(-yln2)d(-yln2)=-e^(xln2)d(xln2)
==>e^(-yln2)=C-e^(xln2) (C是积分常数)
==>2^(-y)=C-2^x
∴原方程的通解是2^(-y)=C-2^x (C是积分常数).