Proof verification: if μ(x,y), v(x,y) are integrating factors of M(x,y)dx+N(x,y)dy=0

[Davidloke]

Proof verification: if μ(x,y), v(x,y) are integrating factors of M(x,y)dx+N(x,y)dy=0

how that if μ(x,y)$ & $v(x,y)are integrating factors of


M(x,y)dx+N(x,y)dy=0


such that μ(x,y)/v(x,y)is not constant! then


μ(x,y)=c(v(x,y)


is a solution to the above differential equation! for every constant c.

My try


μ(x,y)M(x,y)+μ(x,y)N(x,y)=0


If it is a solution to differential equation then


\(\displaystyle \frac{\partial μ(x,y)}{\partial y}{M(x,y)}+μ(x,y)\frac{\partial M(x,y)}{\partial y} \ = \ \frac{\partial μ(x,y)}{\partial x}{N(x,y)}+μ(x,y)\frac{\partial N(x,y)}{\partial x}\)


\(\displaystyle v(x,y)M(x,y)+v(x,y)N(x,y) \ = \ 0\)


For this to be a solution, as for v(x,y)


\(\displaystyle \frac{\partial v(x,y)}{\partial y}{M(x,y)}+v(x,y)\frac{\partial M(x,y)}{\partial y} \ = \ \frac{\partial v(x,y)}{\partial x}{N(x,y)}+v(x,y)\frac{\partial N(x,y)}{\partial x}\)


We can say that


v(x,y)M(x,y)+v(x,y)N(x,y)=μ(x,y)M(x,y)+μ(x,y)N(x,y)


So by ratio we have


\(\displaystyle \frac{μ(x,y)}{v(x,y)} \ = \ c\)

Can someone explain and help me in proof. I just learn differential equation about a week ago like that

 

[Davidloke]

Bump!