令X=x+7,Y=y-4,x+y+3=(x+7)+(y-4)=X+Yx+2y-1=(x+7)+2(y-4)=X+2Y則dy/dx=dY/dX原微分方程化為：dY/dX=(X+Y)/(X+2Y) 齊次方程分子分母同除以X得：dY/dX=(1+Y/X)/(1+2Y/X)令u=Y/X,得Y=uX,dY/dX=u+X*du/dX上式化為：u+X*du/dX=(1+u)/(1+2u)du/dX=(1+u)/(1+2u)-u=(1-2u^2)/(1+2u)則(1+2u)/(1-2u^2)du=dX得：1/(1-2u^2)du+2u/(1-2u^2)du=dX積分得：√2/4*ln|(2u+√2)/(2u-√2)|-(1/2)ln|1-2u^2|=X+C將u換成Y/X得：√2/4*ln|(2Y+√2X)/(2Y-√2X)|-(1/2)ln|1-2(Y/X)^2|=X+C將X換成x+7,Y換成y-4得,兩邊再乘以2：解為√2/2*ln|(2(y-4)+√2(x+7))/(2(y-4)-√2(x+7))|-ln|1-2((y-4)/(x+7))^2|=2x+C1