How to Solve EXACT Differential Equations

Definition of Exact Equation
A differential equation of type
Mdx+Ndy=0 where M and N are all functions of x
is called an exact differential equation if there exists a function of two variables F(x,y) with continuous partial derivatives such that
dF(x,y) = Mdx+Ndy.
The general solution of an exact equation is given by
F(x,y)=C,
where C is an arbitrary constant.
Test for Exactness
Let functions M and N have continuous partial derivatives in a certain domain D. The differential equation Mdx+Ndy =0 is an exact equation if and only if
∂M/∂y=∂N/∂x.
Algorithm for Solving an Exact Differential Equation
First it’s necessary to make sure that the differential equation is exact using the test for exactness:
∂M/∂y=∂N/∂x.
Then we write the system of two differential equations that define the function F(x,y):
∂F/∂x=M and ∂F/∂y=N
Integrate the first equation over the variable x. Instead of the constant C, we write an unknown function of y:
F(x,y) = ∫M(x,y)dx.
The we differentiating with respect to y, we substitute the function F(x,y)into the second equation:
∂F/∂y = ∂/∂y[∫M(x,y)dx+φ(y)]=N(x,y).
From here we get expression for the derivative of the unknown function φ(y):
φ′(y)=M−∂F/∂y
By integrating the last expression, we find the function φ(y) and, hence, the function F(x,y):
F(x,y) = ∫Mdx+φ(y).
The general solution of the exact differential equation is given by
F(x,y)=C.

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[@tambuwalmathsclass]

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[@aribamughal9189]

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[@anaedobechibuikepaschal3507]

Good evening sir please why was 3y*2 not added (8:34) when you found the derivatives of 3xy*2 as you did in the implicit differentiation above

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Manzee

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[@oyedokun]

Determine the most general function M and N such that the equations M(x,y)dx + (2xy³-x⁴y)dy is exact.

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[@sheriffabubakar5944]

good explanation but how can, know if am to solve using implicit only or exact diferential only

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Sir, please show us how to solve exponential in exact differential Equations

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I really don't understand how u got the 3xy²

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