高等数学微分方程问题如图
1)y'-xy'=a(x^2+y')
--->(x+a-1)y'=-ax^2
--->y'=-ax^2/(x+a-1)=ax+(1-a)a+(a-1)^2/(x+a-1)
积分之y=(a/2)x^2+x(a-1)/2+(a-1)^2*ln|x+a-1|+C
2)[e^(x+y)-e^x]dx=[e^(x+y)+e^x]dy
--->e^x*(e^y-1)dx=e^x*(e^y+1)dy
--->dx=[(e^y+1)/(e^y-1]dy
--->dx=[1+2/(e^y-1)]dy
积分之 x+C=y+2ln|1-1/e^y|
【对积分使用代换法，令e^y=t,则dt=e^ydy--->d...全部
1)y'-xy'=a(x^2+y')
--->(x+a-1)y'=-ax^2
--->y'=-ax^2/(x+a-1)=ax+(1-a)a+(a-1)^2/(x+a-1)
积分之y=(a/2)x^2+x(a-1)/2+(a-1)^2*ln|x+a-1|+C
2)[e^(x+y)-e^x]dx=[e^(x+y)+e^x]dy
--->e^x*(e^y-1)dx=e^x*(e^y+1)dy
--->dx=[(e^y+1)/(e^y-1]dy
--->dx=[1+2/(e^y-1)]dy
积分之 x+C=y+2ln|1-1/e^y|
【对积分使用代换法，令e^y=t,则dt=e^ydy--->dy=dt/t
所以2∫dy/(e^y-1)=2∫dt/[t(t-1)]=2∫dt/(t-1)-2∫dt/t
=2ln|t-1)-2ln|t|=2ln|(e^y-1)/e^y|=2ln|1-1/e^y|】
3)(y+1)^2*dy/dx+x^2=0
--->(y+1)^2*dy+x^3*dx=0
两边同时积分之 (1/3)(y+1)^3+(1/4)x^4=C
4(y+1)^3+3x^4=C'。
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