一阶线性微分方程的通解公式(x-2)*dy/dx=y+2*(x-2)^3,求y的通解答案是y=(x-2)^3+C*(x-2),中间有积分问题我不是很懂,

Question

一阶线性微分方程的通解公式
(x-2)*dy/dx=y+2*(x-2)^3,求y的通解
答案是y=(x-2)^3+C*(x-2),中间有积分问题我不是很懂,

Answer 1

∵(x-2)*dy/dx=y+2*(x-2)³
==>(x-2)dy=[y+2*(x-2)³]dx
==>(x-2)dy-ydx=2*(x-2)³dx
==>[(x-2)dy-ydx]/(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)³+C(x-2)
∴原方程的通解是y=(x-2)³+C(x-2) (C是积分常数).