高数微分方程问题求满足初始条件的解xy'+x+sin(x+y)=0,y|(x=π/2)=0(1-cos(x+y))/sin(x+y)=π/2x,

Question

高数微分方程问题
求满足初始条件的解
xy'+x+sin(x+y)=0,y|(x=π/2)=0
(1-cos(x+y))/sin(x+y)=π/2x,

Answer 1

把xy'+x看作是x(x+y)',所以方法就是换元,把因变量从y换成x+y.
令u=x+y,则方程化为x*du/dx+sinu=0.分离变量,cscudu=-dx/x.两边积分,ln(cscu-cotu)=-lnc+lnC,所以cscu-cotu=C/x.代入u=x+y,原方程的通解是csc(x+y)-cot(x+y)=C/x.
由初始条件得C=π/2.
所以特解是csc(x+y)-cot(x+y)=π/2x,或者写成(1-cos(x+y))/sin(x+y)=π/2x,还可以写成tan((x+y)/2)=π/2x,或者y=2arctan(π/2x)-x.