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?

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

Expert

2022-07-06Added 20 answers

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

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