First Order Differential Equations with worked examples
- Examples With Separable Variables Differential Equations
- Linear Type Of Differential Equation
- Equations That Can Be Reduced To The Linear Form
- General Solution Of The Bernoulli Equation
- Homogeneous Equations
- The Method Of Solution For Homogeneous Equations
- The Exceptional Case Of Homogeneous Equations
- Exact Equations
- Page Comments
Examples With Separable Variables Differential EquationsThis article presents some working examples with separable differential equations. Definition Separable Differential Equations are differential equations which respect one of the following forms :
- where F is a two variable function,also continuous.
- , where f and g are two real continuous functions.
- A rational function is a real function respecting where are polynomials.
Example - Simple Differential Equation
As the equation is of first order, integrate the function twice, i.e.
- A trigonometric function is a real function where contains one or more of the trigonometric functions :
Example - Simple Cosine
This is the same as
which we integrate in the normal way to yield
Example - Potential example
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
Substituting in the given values for V and r
Linear Type Of Differential EquationEquations of the type
Where P and Q are function of x ( but not of y) are said to be linear of the first order
Example - Rational equation
If each side of te equation is multiplied by x the equation becomes:-
This equation has been solved by using the obvious integrating factor x. It is possible to find a more general solution by using R as and integrating factor. Consider the following equation :
By Inspection the left hand side of this equation must reduce to (Ry)
This gives the rule that to solve multiply both sides by an integrating factor of:-
Hence the Method of solving this type of equation is :
- Reduce the equation into the form
- Multiply through by the Integrating Factor:-
- The equation becomes :-
Equations That Can Be Reduced To The Linear Form
A linear form in 2 variables is given by where
Analogous for n variables .
Analogous for n variables .
Example - Simple equations
Consider the equation:
Divide through by
This example is a particular case of The Bernoulli Equation
General Solution Of The Bernoulli Equation
Bernoulli equations have an important property :
- they are nonlinear differential equations with known exact solutions.
This section is presenting the Bernoulli Equation.
and are functions of x This can be reduced to a linear form by putting Therefore
The original equation can be re-written as:
Homogeneous EquationsAny equation which can be put into the form:
A homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have the same total degree. For example : is homogenous.
is said to be Homogeneous. To test whether a function of x and y can be written in the form of the right hand side, substitute for . If the result is in the form , i.e. all the x's cancel, then the test is satisfied and the equation is homogeneous.
Example - Testing a function is homogeneous
Is the follow function homogeneous:
Substitute for y=vx,
As all the x have cancelled out, the test is satisfied.
Function is homogeous
The Method Of Solution For Homogeneous EquationsSubstitute in both sides of the equation
Note. If y is a function of x then so is v
Thus the equatican be re-written as:
Re-writing and Separating the variables:
Example - Homogenous
Putting y = vx
Substituting for v Therefore
The Exceptional Case Of Homogeneous EquationsIf the straight lines are parallel there is no finite point of intersection and the method of solving such equations is illustrated by the following example.
Put Z = 3y - 4x and thus The equation can now be written as:
Replacing Z the solution to the differential equation is :
A form is said to be exact in a region if there is a function such as .
is an exact differential.
Thus the equation giving that i.e. is called an exact Equation.
Example - Exact differential
This equation is not exact as it stands but if it is multiplied through by it becomes: