Separable
This section contains worked examples of the type of differential equation which can be solved by integration
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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 :
- \inline \displaystyle \frac{dy}{dx} = F(x,y) where \inline F is a two variable function, also continuous.
- \inline \displaystyle f(y)dy=g(x)dx, where \inline f and \inline g are two real continuous functions.
Rational Functions
- A rational function on \inline \mathhf{R} is a function \inline f:\mathhf{R}\to\mathhf{R} which can be expressed as \inline \displaystyle f(x)=\frac{P(x)}{Q(x)} where \inline P,Q 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
Solution
Trigonometric Functions
- A rational function on \inline \mathhf{R} is a function \inline f:\mathhf{R}\to\mathhf{R} which can be expressed as a combination of trigonometric functions (\inline sinx,cosx,tanx,cotanx).
Example:
Example - Simple Cosine
Problem
Workings
This is the same as
which we integrate in the normal way to yield
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
Substituting in the given values for V and r
and
Thus
Solution