解微分方程组dx/dt=2x-4y,dy/dt=-5x+3y

Question

解微分方程组dx/dt=2x-4y,dy/dt=-5x+3y

Answer 1

dx/dt=2x-4y
dy/dt=-5x+3y
dy/dx=(-5x+3y)/(2x-4y)
(2x-4y)dy=(-5x+3y)dx
2xdy-3ydx=4ydy-5xdx
y=xu,dy=xdu+udx
2x(xdu+udx)-3xudx=4xu(xdu+udx)-5xdx
2xdu+2udx-3udx=4uxdu+4u^2dx-5dx
2x(1-2u)du=(4u^2+u-5)dx
(4u-2)du/[4u^2+u-5]=-dx/(x)
(1/18)ln|4u-5|+(28/9)ln|u-1|=C1-ln|x|
通解
(1/18)ln|4y/x -5| +(28/9)ln|y/x-1|=C1-ln|x|
∫(4u-2)du/(4u^2+u-5)=∫(4u-2)du/[(4u+5)(u-1)]=∫(2/9)du/(4u-5)+∫(28/9)du/(u-1)
=(1/18)ln|4u-5|+(28/9)ln|u-1|
(1/18)ln|4y-5x|+(28/9)ln|y-x|-(1/18)lnx-(28/9)lnx=C1-lnx