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)