Separable
This section contains worked examples of the type of differential equation which can be solved by integration
Separable Differential Equations
This section contains worked examples of the type of differential equation which can be solved by direct Integration.Definition
- Separable Differential Equations are differential equations which respect one of the following forms :
where
is a two variable function, also continuous.
, where
and
are two real continuous functions.
Rational Functions
- A rational function on
is a function
which can be expressed as
where
are two polynomials.
Example:
Example - Simple Differential Equation
Problem
Solve:
Workings
As the equation is of first order, integrate the function twice, i.e.
and
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Solution
Trigonometric Functions
- A rational function on
is a function
which can be expressed as a combination of trigonometric functions (
).
Example:
Example - Simple Cosine
Problem
Workings
This is the same as
which we integrate in the normal way to yield
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Solution
Physics Examples
Example:
Example - Potential example
Problem
If a and b are the radii of concentric spherical conductors at potentials of
respectively, then V is the potential at a distance r from the centre. Find the value of V if:
and
at r=a and
at r=b
Workings
Solution