Question

求解常微分方程(dx/dt)((t^2)(x^3)+tx)=1

rt,求大神解答，试着想凑成全微分形式但是怎么都不成功...

Answer 1

→dt/dx=(t^2)(x^3)+tx
→(1/t^2)(dt/dx)=x^3+x/t
令u=1/t,则du/dt=-1/t^2
即 -du/dx=x^3+ux
写成一阶微分方程的一般形式为
u'(x)+x·u=-x^3
其通解为
u=e^(-∫xdx)·[-∫x^3·e^(∫xdx) dx + C]
=e^(-x²/2)·[-∫x^3·e^(x²/2) dx + C]
=e^(-x²/2)·[-(1/2)∫x^2·e^(x²/2) dx² + C]
=e^(-x²/2)·[-∫x^2·e^(x²/2) d(x²/2) + C]
=e^(-x²/2)·[-∫x^2 d(e^(x²/2)) + C]
=e^(-x²/2)·[-x^2·(e^(x²/2)) + 2∫(e^(x²/2)) d(x²/2) + C]
=e^(-x²/2)·[-x^2·(e^(x²/2)) + 2(e^(x²/2)) + C]
= -x² +2 +C/e^(x²/2)
即
1/t= -x² +2 +C/e^(x²/2)