æ±ä¸åå¾®åæ¹ç¨ç解
(1) .(x+y)dy+(x-y)dx=0
解ï¼(x+y)dy=(y-x)dxï¼æ
dy/dx=(y-x)/(y+x)=(y/x-1)/(y/x+1)...........(1)ï¼
令y/x=uï¼å³y=uxï¼å 为dy/dx=u+xdu/dxï¼äºæ¯æ¹ç¨(1)å为ï¼
u+xdu/dx=(u-1)/(u+1)ï¼ä¹å°±æ¯xdu/dx=(u-1)/(u+1)-u=-(u²+1)/(u+1)ï¼
å离åéå¾[(u+1)/(u²+1)]du=-(1/x)dxï¼å³æudu/(u²+1)+du/(u²+1)=-(1/x)dxï¼
ä¹å°±æ¯æ(1/2)d(u²+1)/(u²+1)=-(1/x)dxï¼
积åä¹å¾(1/2)ln(u²+1)=-lnx+lnC₁=ln(C₁/x)
äºæ¯å¾â(u²+1)=C₁/xï¼å°u=y/x代å
¥å³å¾â[(y/x)²+1]=(1/x)â(x²+y²)=C₁/x
åç®å¾â(x²+y²)=C₁ï¼æ
åæ¹ç¨çé解为x²+y²=Cï¼å
¶ä¸C=C²₁
(2)ylnydx+(x-lny)dy=0
解ï¼dx/dy=(lny-x)/ylny=1/y-x/ylny
å³ædx/dy+x/ylny=1/y............(1)
为äºæ±(1)ç解ï¼å
èèæ¹ç¨ï¼dx/dy+x/ylny=0..........(2)
å°æ¹ç¨(2)å离åéï¼å¾dx/x=-dy/ylny=-d(lny)/lny
积åä¹å¾lnx=-lnlny+lnC₁=ln(C₁/lny)
æ
å¾x=C₁/lnyï¼å°ä»»æ常æ°C₁æ¢æyçå½æ°uï¼å³æx=u/lny..........(3)ï¼
æ
dx/dy=[(lny)(du/dy)-u/y]/(lny)²=(du/dy)(1/lny)-u/(yln²y).........(3â²)
å°(3)å(3â²)代å
¥(1)å¼å¾ï¼
(du/dy)(1/lny)-u/(yln²y)+u/(yln²y)=1/y
äºæ¯å¾(du/dy)(1/lny)=1/y
å离åéå¾du=lnydy/y=lnyd(lny)
积åä¹å¾u=(1/2)ln²y+Cï¼ä»£å
¥(3)å¼å³å¾x=[(1/2)ln²y+C]/lnyï¼è¿å°±æ¯åæ¹ç¨çé解ã
(3) y'=(2-x+y)²
解ï¼dy/dx=4+x²+y²-4x+4y-2xy
dy/dx-y²-2(2-x)y=(2-x)²............(1)
å
èèæ¹ç¨dy/dx-y²-2(2-x)y=0
(4)y''=3*y^(1/2)
解ï¼
追é®åçæç¹å°é®é¢ï¼ä¸è¿æè·¯å¾å¯¹ï¼ä½éè¦çæ¯å2个
追ç(4)y''=3y^(1/2)
解ï¼è®¾yâ²=dy/dx=pï¼åyâ²â²=dp/dx=(dp/dy)(dy/dx)=p(dp/dy)
代å
¥åå¼å¾p(dp/dy)=3y^(1/2)
å离åéå¾pdp=3[y^(1/2)]dy
积åä¹å¾(1/2)p²=2y^(3/2)+(1/2)C₁
p²=4y^(3/2)+C₁
æ
p=â[4y^(3/2)+C₁]
äºæ¯å¾dy/dx=â[4y^(3/2)+C₁]
dy/â[4y^(3/2)+C₁]=dx
ä¸é¢çé®é¢å°±æ¯æ±è§£è¿ä¸ªç§¯åã
(3) y'=(2-x+y)²
解ï¼ä»¤2-x+y=zï¼ä¸¤è¾¹å¯¹xå导æ°å¾-1+dy/dx=dz/dxï¼å³ædy/dx=1+dz/dxï¼ä»£å
¥åå¼å¾ï¼
1+dz/dx=z²ï¼å³ædz/dx=z²-1ï¼
æ
dz/(z²-1)=dxï¼ç§¯åä¹å¾â«dz/(z²-1)=(1/2)â«[1/(z-1)-1/(z+1)]dz
=(1/2)[ln(z-1)-ln(z+1)]=x+(1/2)C
å³æln[(z-1)/(z+1)=2x+Cï¼å°z=2-x+y代å
¥å³å¾åæ¹ç¨çé解为ï¼
ln[(1-x+y)/(3-x+y)]=2x+C.
追é®ï¼3ï¼å¦ä½ç§¯å
追ç(4)y''=3y^(1/2)
解ï¼è®¾yâ²=dy/dx=pï¼åyâ²â²=dp/dx=(dp/dy)(dy/dx)=p(dp/dy)
代å
¥åå¼å¾p(dp/dy)=3y^(1/2)ï¼å离åéå¾pdp=3[y^(1/2)]dyï¼ç§¯åä¹å¾(1/2)p²=2y^(3/2)+2C₁
p²=4y^(3/2)+4C₁ï¼æ
p=2â[y^(3/2)+C₁]
äºæ¯å¾dy/dx=2â[y^(3/2)+C₁]ï¼dy/â[y^(3/2)+C₁]=2dxï¼
令y=u²ï¼ådy=2uduï¼ä»£å
¥ä¸å¼å¾ï¼2â«udu/â(u³+C₁)=2X+C₂ï¼
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