求微分方程(x-2)dy/dx=y+2(x-3)^3的通解，在线等
2019-05-28
求微分方程(x-2)dy/dx=y+2(x-3)^3的通解，在线等
优质解答
∵(x-2)dy/dx=y+2(x-3)^3
==>(x-2)dy-ydx=2(x-3)^3dx
==>((x-2)dy-ydx)/(x-2)^2=2(x-3)^3dx/(x-2)^2 (等式两端同除(x-2)^2)
==>d(y/(x-2))=(2x-10+6/(x-2)-2/(x-2)^2)dx
==>∫d(y/(x-2))=∫(2x-10+6/(x-2)-2/(x-2)^2)dx
==>y/(x-2)=x^2-10x+6ln│x-2│+2/(x-2)+C (C是积分常数)
==>y=(x-2)(x^2-10x)+6(x-2)ln│x-2│+2+C(x-2)
==>y=x^3-12x^2+20x+2+6(x-2)ln│x-2│+C(x-2)
∴原方程的通解是y=x^3-12x^2+20x+2+6(x-2)ln│x-2│+C(x-2)。
∵(x-2)dy/dx=y+2(x-3)^3
==>(x-2)dy-ydx=2(x-3)^3dx
==>((x-2)dy-ydx)/(x-2)^2=2(x-3)^3dx/(x-2)^2 (等式两端同除(x-2)^2)
==>d(y/(x-2))=(2x-10+6/(x-2)-2/(x-2)^2)dx
==>∫d(y/(x-2))=∫(2x-10+6/(x-2)-2/(x-2)^2)dx
==>y/(x-2)=x^2-10x+6ln│x-2│+2/(x-2)+C (C是积分常数)
==>y=(x-2)(x^2-10x)+6(x-2)ln│x-2│+2+C(x-2)
==>y=x^3-12x^2+20x+2+6(x-2)ln│x-2│+C(x-2)
∴原方程的通解是y=x^3-12x^2+20x+2+6(x-2)ln│x-2│+C(x-2)。