My question isIf u(x,y) and v(x,y) are two integrating factors of a diff eqn M(x,y)dx+N(x,y)dy, u/v is not a constant. then u(x,y)=cv(x,y)is a solution to the differential eqn for every constant c. I m totally stuck :(Another doubt i have is how to derive the singular solution for the Clairaut's equation. i tried it we have y=px+f(p) diff wrt x and considering dp/dx=0 we get p=c, how to solve the other part?

Question

My question isIf u(x,y) and v(x,y) are two integrating factors of a diff eqn M(x,y)dx+N(x,y)dy, u/v is not a constant. then u(x,y)=cv(x,y)is a solution to the differential eqn for every constant c. I m totally stuck :(Another doubt i have is how to derive the singular solution for the Clairaut&#039;s equation. i tried it we have y=px+f(p) diff wrt x and considering dp/dx=0 we get p=c, how to solve the other part?

Caiden Barrett · Accepted Answer

The fact that u is an integrating factor means that   (  u  M      )    y    =  (  u  N      )    x  , i.e.       u    x    =            u      (              M        y            −              N        x            )      +              u        y            M        N  , and similarly       v    x    =            v      (              M        y            −              N        x            )      +              v        y            M        N  . So   (  u      /    v      )    x    =                    u        x            v      −      u              v        x                    v      2        =            (              u        y            v      −      u              v        y            )      M              N              v        2              =      M    N    (  u      /    v      )    y  . The curves u/v=c satisfy the differential equation   (  u      /    v      )    x       d  x  +  (  u      /    v      )    y       d  y  =  0 which is a multiple of M dx+N dy=0.