solve for general solution [x csc(y/x) -y] dx +xdy = 0
W warsatan New member Joined Sep 12, 2005 Messages 36 Sep 4, 2006 #1 solve for general solution [x csc(y/x) -y] dx +xdy = 0
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Sep 5, 2006 #2 Hello, warsatan! The equation is homogenous . . . Solve for general solution: \(\displaystyle \L\,\left[x\csc\left(\frac{y}{x}\right)\,-\,y\right]dx\,+\,x\,dy \;= \;0\) Click to expand... We have: \(\displaystyle \L\,x\,dy \;= \;\left[y\,-\,x\csc\left(\frac{y}{x}\right)\right]\,dx\;\;\Rightarrow\;\;\frac{dy}{dx}\;=\;\left[\frac{y}{x}\,-\,\csc\left(\frac{y}{x}\right)\right]\) Let \(\displaystyle v\,=\,\frac{y}{x}\;\;\Rightarrow\;\;y\,=\,xv\;\;\Rightarrow\;\;\frac{dy}{dx}\:=\:x\frac{dv}{dx}\,+\,v\) Substitute: \(\displaystyle \L\,x\frac{dv}{dx}\,+\,v\;=\;v\,-\,\csc v\) Then: \(\displaystyle \L\,x\frac{dv}{dx}\:=\:-\csc v\;\;\Rightarrow\;\;\frac{dv}{\csc v}\:=\:-\frac{dx}{x}\;\;\Rightarrow\;\;\sin v\,dv\:=\:-\frac{dx}{x}\) Can you finish it now?
Hello, warsatan! The equation is homogenous . . . Solve for general solution: \(\displaystyle \L\,\left[x\csc\left(\frac{y}{x}\right)\,-\,y\right]dx\,+\,x\,dy \;= \;0\) Click to expand... We have: \(\displaystyle \L\,x\,dy \;= \;\left[y\,-\,x\csc\left(\frac{y}{x}\right)\right]\,dx\;\;\Rightarrow\;\;\frac{dy}{dx}\;=\;\left[\frac{y}{x}\,-\,\csc\left(\frac{y}{x}\right)\right]\) Let \(\displaystyle v\,=\,\frac{y}{x}\;\;\Rightarrow\;\;y\,=\,xv\;\;\Rightarrow\;\;\frac{dy}{dx}\:=\:x\frac{dv}{dx}\,+\,v\) Substitute: \(\displaystyle \L\,x\frac{dv}{dx}\,+\,v\;=\;v\,-\,\csc v\) Then: \(\displaystyle \L\,x\frac{dv}{dx}\:=\:-\csc v\;\;\Rightarrow\;\;\frac{dv}{\csc v}\:=\:-\frac{dx}{x}\;\;\Rightarrow\;\;\sin v\,dv\:=\:-\frac{dx}{x}\) Can you finish it now?
