求微分方程通解 d^2y/dx^2-e^y* dy/dx=0
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求微分方程通解 d^2y/dx^2-e^y* dy/dx=0
![求微分方程通解 d^2y/dx^2-e^y* dy/dx=0](/uploads/image/z/7997370-42-0.jpg?t=%E6%B1%82%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E9%80%9A%E8%A7%A3+d%5E2y%EF%BC%8Fdx%5E2-e%5Ey%2A+dy%EF%BC%8Fdx%3D0)
令p=dy/dx, 则d^2y/dx^2=pdp/dy
代入方程:pdp/dy-e^yp=0
dp/dy=e^y
dp=e^ydy
积分:p=e^y+c
dy/dx=e^y+c
dy/(e^y+c)=dx
d(e^y)/[e^y(e^y+c)]=dx
d(e^y)[1/e^y-1/(e^y+c)]=cdx
积分:lne^y/(e^y+c)=cx+c1
e^y/(e^y+c)=c1e^(cx)
解得:
y=ln{cc1e^(cx)/[1-c1e^(cx)]}
代入方程:pdp/dy-e^yp=0
dp/dy=e^y
dp=e^ydy
积分:p=e^y+c
dy/dx=e^y+c
dy/(e^y+c)=dx
d(e^y)/[e^y(e^y+c)]=dx
d(e^y)[1/e^y-1/(e^y+c)]=cdx
积分:lne^y/(e^y+c)=cx+c1
e^y/(e^y+c)=c1e^(cx)
解得:
y=ln{cc1e^(cx)/[1-c1e^(cx)]}
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