Differential Equations


   
 
Some Methods of Solving First Order First Degree Differential Equation
The different ways of solving differential equation are a follows:
 
Method of separation of variables
 
Homogeneous differential equations
 
Linear differential equations
 
Method of Separation of Variables
 
Definition 7
 
A first - order first - degree differential equation is of the form
 
 
If the function f turns out to be of the form
 
f(x, y) = p(x) q(y) ….(2)
 
(i.e., the product of a function of x with a function of y, then the differential equation (1) is said to have the variable x, y separable.)
 
In this case the differential equation (1) can be written in the alternative form as
 
 
If P(x) is a primitive of p(x), and Q(y) is a primitive of , then (3) implies
 
Q(y) - P(x) = C (real number) ….(4)
 
This is expressed in integral notation as
 
 
We say that (5) is obtained from (3) by integration. Thus, (5) is a primitive of differential equation (1) with f(x, y) given by (2).
 
Note:
 
The word 'primitive' has been used both in reference to a differential equation, and in reference to a function, but there should be no confusion. By a primitive of a given function , we mean a function , such that g'(x) = f(x) for all xR. In other words, y = g(x), (xI) is a primitive of the differential equation y' = f(x), (xI).
 
A differential equation is said to be of the variables separable type if it can be in rearranged the form f(x)dx=g(y)dy whose solution is obtained by direct integration.
 
 
 
 
 
Example:
 
Solve the differential equation
 
 
Suggested answer:
 
 
 
By integration
 
 
y = tan-1x+c
 
Homogeneous Differential Equation
 
A differential equation is said to be a homogeneous differential equation. If it is of the form
 
 
where f and g are homogenous functions, in x and y of the same degree.
 
Note:
 
A function f(x,y) is said to be homogenous of degree n if it can be expressed in the form

 

 
Definition 8:
 
The first-order first-degree differential equation
 
 
is said to be homogeneous, if f is a homogeneous function of degree zero,
 
,
 
In other words, f instead of depending on x and y separately, depends on . Thus,
 
 
Examples:
 
 
 
A homogeneous differential equation is solved by using y = vx and transforming the given differential equation to a variables separable form in x and v.
 
To solve homogeneous differential equation we use the substitution.
 
y = vx
 
 
The equation reduces to variables separable.
 
Example:
 
Solve
 
 
Suggested answer:
 
 
Let y = vx
 
 
 
 
 
v = log x +c
 
Resubstituting for v, we get
 
 
Note:
 
A differential equation is said to be linear when the dependent variable and its derivatives occur in the first degree and are not multiplied together.
 
 
When f1 to fn as the functions of x, in general linear differential equations of order n.
 
First Order Linear differential equations
 
Definition 9:
 
A first-order differential equation is said to be linear if, in it, the unknown function y and its derivative y' appear with non-negative integral index not greater than one and not as product yy' either.
 
Hence, a first-order linear differential equation is of the form:
 
 
where p and q are given real-valued function on I. The differential equation
 
 
is also a linear differential equation, where y is the independent variable and x is the dependent variable.
 
Let us define for arbitrary chosen a Î I. The term eP(x) is called an integrating factor, (I.F.), of the differential equation(1) since
 
 
which is easily integrable. Integrating (2), we obtain
 
, (C : parameter) ...(3)
 
The relation (3) gives a primitive of the differential equation (1).
 
A Leibnitz's linear differential equation is a linear differential of first order which is of the form.
 
 
P and Q are the functions of x only.
 
Its solution is given by
 
where is called the integrating factor of the differential equation.
 
Note:
 
 
Where P and Q are functions of y only is a Leibnitz's linear differential equation which is linear in x and its solution is given by
 
 
C is a constant of integration.
 
Example:
 
Solve
 
 
Suggested answer:
 
Divide by x
 
 
 
 
 
The general solution is
 
 
 
 
 
Important points
 
Degree and order of a differential equation
 
How to form a differential equation
 
Types of differential equations and their general solutions.
 
  
     
   
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