大一高数求微分方程的通解

Question

如题所述

Answer 1

2.解：∵(x-2)dy/dx=y+2(x-2)^3
==>(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是常数)
==>y=(x-2)^3+C(x-2)
∴原方程的通解是y=(x-2)^3+C(x-2)。
3.解：令x=ty，则dx=tdy+ydt
代入原方程，化简得 (t+2e^t)dy+y(1+2e^t)dt=0
==>dy/y+(1+2e^t)dt/(t+2e^t)=0
==>d(ln│y│)+d(ln│t+2e^t│)=0
==>ln│y│+ln│t+2e^t│=ln│C│ (C是常数)
==>y(t+2e^t)=C
==>y(x/y+2e^(x/y))=C
==>x+2ye^(x/y)=C
故原方程的通解是x+2ye^(x/y)=C。