Exact differential equation





Exact differential equation

A differential equation is a equation used to define a relationship between a function and derivatives of that function. Differential equation is extremely used in the field of engineering, physics, economics and other disciplines.

 

A differential equation of the form Mdx + N dy = 0, where M & N are function of x & y, is called exact if there exists a function f(x, y) such that Mdx + Ndy = d f (x, y).

 

Note: a necessary and sufficient condition for the differential equation Mdx + Ndy = 0 to be exact is  \dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}

 

Some important relations

 

1. x dy = y dx = d (x, y) \\ 2. x dx + y dy = \dfrac{1}{2} d (x^2 + y^2) \\ 3. \dfrac{x dy- y dx}{x^2} = d\left(\dfrac{y}{x} \right) \\ 4. \dfrac{y dx- y dy}{y^2} = d\left(\dfrac{x}{y} \right) \\ 5. \dfrac{x dy- y dx}{xy} = \dfrac{dy}{y}- \dfrac{dx}{x} = d \log \left( \dfrac{y}{x} \right) \\ 6. \dfrac{2xy dx- x^2 dy}{y^2} = d \left(\dfrac{x^2}{y} \right) \\ 7. \dfrac{2xy dy- y^2 dx}{x^2} = d\left(\dfrac{y^2}{x} \right) \\ 8. \dfrac{y dx- xdy}{x^2 + y^2} = d\left(\tan^{-1}\frac{x}{y} \right) \\ 9. \dfrac{x dy- y dx}{x^2 + y^2} = d\left( \tan^{-1}\frac{y}{x} \right)

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