∵(x-2)dy=[y+2(x-2)³]dx
==>(x-2)dy-ydx=2(x-2)³dx
==>[(x-2)dy-yd(x-2)]/(x-2)²=2(x-2)dx
==>d[y/(x-2)]=d[(x-2)²]
==>y/(x-2)=(x-2)²+C (C是积分常数)
==>y=(x-2)[(x-2)²+C]
∴原方程的通解是y=(x-2)³+C(x-2).
求方程的通解 (x-2)dy=[y+2(x-2)^3]dx 貌似书本上答案错了?
∵(x-2)dy=[y+2(x-2)³]dx
==>(x-2)dy-ydx=2(x-2)³dx
==>[(x-2)dy-yd(x-2)]/(x-2)²=2(x-2)dx
==>d[y/(x-2)]=d[(x-2)²]
==>y/(x-2)=(x-2)²+C (C是积分常数)
==>y=(x-2)[(x-2)²+C]
∴原方程的通解是y=(x-2)³+C(x-2).