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How to solve first Order Differential Equation | lesson 2
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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.
If you find this video interesting, kindly subscribe to my channel for more exciting Maths tutorials.
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.
If you find this video interesting, kindly subscribe to my channel for more exciting Maths tutorials.
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