2d3vljtq

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?

Caiden Barrett

Expert

The fact that u is an integrating factor means that $\left(uM{\right)}_{y}=\left(uN{\right)}_{x}$, i.e. ${u}_{x}=\frac{u\left({M}_{y}-{N}_{x}\right)+{u}_{y}M}{N}$, and similarly ${v}_{x}=\frac{v\left({M}_{y}-{N}_{x}\right)+{v}_{y}M}{N}$. So $\left(u/v{\right)}_{x}=\frac{{u}_{x}v-u{v}_{x}}{{v}^{2}}=\frac{\left({u}_{y}v-u{v}_{y}\right)M}{N{v}^{2}}=\frac{M}{N}\left(u/v{\right)}_{y}$. The curves u/v=c satisfy the differential equation which is a multiple of M dx+N dy=0.