Question

第四题怎么解啊

Answer 1

求微分方程y''=1+y'²的通解解：令y'=dy/dx=p；则y''=dy'/dx=(dy'/dy)(dy/dx)=p(dp/dy);代入原式得p(dp/dy)=1+p²;分离变量得：pdp/(1+p²)=dy积分之得：∫pdp/(1+p²)=y+lnc₁(1/2)∫d(1+p²)/(1+p²)=y+lnc₁ln(1+p²)=2y+2lnc₁；1+p²=e^(2y+lnc₁²)=c₁²e^(2y)=(c₁e^y)²;故p²=(c₁e^y)²-1;∴p=dy/dx=±√[(c₁e^y)²-1];再分离变量得dy/√[(c₁e^y)²-1]=±dx..①令c₁e^y=secu，则e^y=(1/c₁)secu.②；e^ydy=(1/c₁)secutanudu;dy=[(1/c₁)secutanudu]/e^y={[(1/c₁)secutanu]/(1/c₁)secu}du=tanudu;代入①式并化简得：du=±dx；积分之得u=±x+c₂③；将③代入②得：e^y=(1/c₁)sec(±x+c₂)故原方程的通解为：y=ln[(1/c₁)sec(±x+c₂)]

Answer 2

4.
x²-3x-4≤-4
x(x-3)≤0
0≤x≤3
y=x²-3x-4=(x- 3/2)²-25/4
对称轴x=3/2
最小值-25/4，x=3/2在区间上，
m≥3/2
综上，得：3/2≤m≤3
选C