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,...

2d3vljtq

2d3vljtq

Answered

2022-07-05

My question is
If 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?

Answer & Explanation

Caiden Barrett

Caiden Barrett

Expert

2022-07-06Added 20 answers

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.

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