求齐次方程dy/dx=-(4x 3y)/(x y)的通解

求齐次方程dy/dx=-(4x 3y)/(x y)的通解

解：∵令t=lny，则y=e^t，dy=e^tdt ∴代入原方程得 dy/dx=y/(x+y^4) ==>ydx/dy=(x+y^4) ==>e^tdx/(e^tdt)=(x+e^(4t)) ==>dx/dt=x+e^(4t)..........(1) ∵很容易求出齐次方程dx/dt=x的通解是 x=Ce^t (C是积分常数) ∴根据常数变易法，设方程(1)的解为x=C(t)e^t (C(t)是关于t的函数) ∵代入方程(1)得 C'(t)e^t+C(t)e^t=C(t)e^t+e^(4t) ==>C'(t)e^t=e^(4t) ==>C'(t)=e^(3t) ==>C(t)=e^(3t)/3+C (C是积分常数) ∴方程(1)的通解是x=(e^(3t)/3+C)e^t=e^(4t)/3+Ce^t 故原方程的通解是x=y^4/3+Cy (C是积分常数)。

dy/dx=-(4x+3y)/(x+y)=-(4+3y/x)/(1+y/x) 令u=y/x，则y=xu，dy/dx=u+xdu/dx u+xdu/dx=-(4+3u)/(1+u) xdu/dx=-(4+4u+u^2)/(1+u)=-(u+2)^2/(u+1) (u+1)/(u+2)^2du=-dx/x ∫[1/(u+2)-1/(u+2)^2]du=∫-dx/x ln|u+2|+1/(u+2)=-ln|x|+c (u+2)e^[1/(u+2)]=c/x (y/x+2)e^[1/(y/x+2)]=c/x (y+2x)e^[x/(y+2x)]=c，其中c是任意常数