Question

3. If M(x, y)dx + N(x, y)dy = 0 is a homogeneous differential equation then the
integrating factor of M dx + N dy = 0 is​

Answer 1

Answer:

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Answer 2

If $M(x, y)dx + N(x, y)d y = 0$ is homogeneous differential equation with $Mx + N y \neq 0$, then integrating factor will be $\frac{1}{Mx+ N y}$

• A function f(x, y) is  such that $f(tx,ty) = t^{n} f(x,y)$ then the function is homogeneous of degree n,  for all given x, y and t.

• A differential equation of the form

• $\frac{dy}{dx} = f(\frac{y}{x} ) or \frac{dy}{dx} = f(\frac{P(x,y)}{Q(x,y)} )$ said to be homogeneous differential equation, here P(x,y) and Q(x,y) are homogeneous functions.

• If a differential equation having the form $M(x,y)dx+N(x,y)dy=0$, does not satisfy condition  $\frac{dM}{dy} \neq \frac{dN}{dx}$ , then the differential equation is not exact.
• Such a differential equation can be made exact by multiplying it by a function f (x, y), such that $f (x, y)[M(x, y)dx +N(x, y)dy ] = 0$ , then f (x, y) is called an integrating factor.

• If $M(x, y)dx + N(x, y)d y = 0$ is homogeneous differential equation with $Mx + N y \neq 0$, then integrating factor will be $\frac{1}{Mx+ N y}$

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