求解下列微分方程 ①dy/dx=(x+y)/(x-y)②(x-y)ydx-x^2dy=0③dy/dt+ytant=sin
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求解下列微分方程 ①dy/dx=(x+y)/(x-y)②(x-y)ydx-x^2dy=0③dy/dt+ytant=sin2t④ylnydx+(x-lny)dy=0
![求解下列微分方程 ①dy/dx=(x+y)/(x-y)②(x-y)ydx-x^2dy=0③dy/dt+ytant=sin](/uploads/image/z/8037023-23-3.jpg?t=%E6%B1%82%E8%A7%A3%E4%B8%8B%E5%88%97%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B+%E2%91%A0dy%2Fdx%3D%28x%2By%29%2F%28x-y%29%E2%91%A1%EF%BC%88x-y%29ydx-x%5E2dy%3D0%E2%91%A2dy%2Fdt%2Bytant%3Dsin)
1
dy/dx=(x+y)/(x-y)
y=xu
dy=xdu+udx
xdu+udx=(1+u)/(1-u)dx
xdu=[(1+u)/(1-u)-u]dx
(1-u)du/(1+u^2)=dx/x
arctanu-ln|u|=ln|x|-lnC
arctanu+lnC=ln|y|
y=Ce^(arctan(y/x)通解
2
(xy-y^2)dx-x^2dy=0
y=xu
dy=xdu+udx
(x^2u-x^2u^2)dx-x^2(xdu+udx)=0
(-u^2)dx-xdu=0
dx/x=du/(-u^2)
ln|x|+lnC=1/u
通解Cx=e^(x/y)
3
dy/dt+ytant=sin2t=2sintcost
dy=(2sintcost-ytant)dt
costdy=2sintcostdt+ydcost
(costdy-ydcost)/cost^2=2tantdt
通解y/cost=-2ln|cost|+lnC
4
ylnydx+(x-lny)dy=0
y=e^t
ylny=te^t
te^tdx+(x-t)e^tdt=0
tdx+(x-t)dt=0
tdx+xdt=tdt
xt=(1/2)t^2+C
通解xlny=(1/2)(lny)^2+C
dy/dx=(x+y)/(x-y)
y=xu
dy=xdu+udx
xdu+udx=(1+u)/(1-u)dx
xdu=[(1+u)/(1-u)-u]dx
(1-u)du/(1+u^2)=dx/x
arctanu-ln|u|=ln|x|-lnC
arctanu+lnC=ln|y|
y=Ce^(arctan(y/x)通解
2
(xy-y^2)dx-x^2dy=0
y=xu
dy=xdu+udx
(x^2u-x^2u^2)dx-x^2(xdu+udx)=0
(-u^2)dx-xdu=0
dx/x=du/(-u^2)
ln|x|+lnC=1/u
通解Cx=e^(x/y)
3
dy/dt+ytant=sin2t=2sintcost
dy=(2sintcost-ytant)dt
costdy=2sintcostdt+ydcost
(costdy-ydcost)/cost^2=2tantdt
通解y/cost=-2ln|cost|+lnC
4
ylnydx+(x-lny)dy=0
y=e^t
ylny=te^t
te^tdx+(x-t)e^tdt=0
tdx+(x-t)dt=0
tdx+xdt=tdt
xt=(1/2)t^2+C
通解xlny=(1/2)(lny)^2+C
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