求微分方程 .dy/dx-3xy=x 的通解.

题目

求微分方程 .dy/dx-3xy=x 的通解.

答案

解法一：∵dy/dx-3xy=x ==>dy/dx=x(3y+1)
==>dy/(3y+1)=xdx
==>ln│3y+1│=3x²/2+ln│3C│ (C是积分常数)
==>3y+1=3Ce^(3x²/2)
==>y=Ce^(3x²/2)-1/3
∴原微分方程的通解是y=Ce^(3x²/2)-1/3 (C是积分常数).
解法二：∵dy/dx-3xy==0 ==>dy/y=3xdx
==>ln│y│=3x²/2+ln│C│ (C是积分常数)
==>y=Ce^(3x²/2)
∴根据常数变易法,设原方程得解为y=C(x)e^(3x²/2) (C(x)表示关于x的函数)
∵y'=C'(x)e^(3x²/2)+3xC(x)e^(3x²/2)
代入原方程,得C'(x)e^(3x²/2)+3xC(x)e^(3x²/2)-3xC(x)e^(3x²/2)=x
==>C'(x)e^(3x²/2)=x
==>C'(x)=xe^(-3x²/2)
∴C(x)=∫xe^(-3x²/2)dx
=(1/3)∫e^(-3x²/2)d(3x²/2)
=C-e^(-3x²/2)/3 (C是积分常数)
==>y=C(x)e^(3x²/2)=(C-e^(-3x²/2)/3)e^(3x²/2)=Ce^(3x²/2)-1/3
故原微分方程的通解是y=Ce^(3x²/2)-1/3 (C是积分常数).