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ORDINARY DIFFERENTIAL EQUATIONS (ODEs)
Introduction
Definition1.
A differential equation is an equation that involves derivatives or differentials.
Example: The equations
1.
2. and 3. (^ )^ Are differential equations.
Definition2.
A differential equation that involves only one independent variable is called an ordinary differential equation (ODE).
Example: The equations
1. and 2. (^ )^ are ordinary differential equations.
Definition3.
The order of a differential equation is the order of the highest derivative which appears.
Definition4.
The degree of a differential equation is the order of the highest ordered derivative which appears.
Example: Find the order and the degree of the following D.Es:
1. The equation is an equation of the first order, and first degree. 2. The equation ( ) ( ) is an equation of the second order, and third degree.
Definition5.
The solution or integral of a differential equation is any function, ( )^ which, when put into the equation convents it into an identity.
Example: Find the differential equation associated with
Solution:
Differentiate both sides with respect to x, we get
Which, can be written as
Example: Find the differential equation associated with
Solution:
Or ( ) Or
Example: Find the differential equation associated with the primitive
( ) (^) ( )
Solution:
( ) ( )
( ) ( )
Adding ( ) and( ), we get
Example: Find the differential equation associated with the primitive
( )
Solution:
( ) ( )
Multiplying ( ) by 2 and adding it to (^ )
( ) ( )
We get
( )
Multiplying ( ) by 2 and adding it to ( )
( ) ( )
We get
( )
Multiplying ( ) by 3 and abstracting it from ( )
( ) ( )
We get
Exercises:
Find the order and the degree of the following D.Es:
1. ( ) (^ )^ 2. √
4. (^ )
Find the differential equation associated with:
1. 2.
Find the differential equation associated with the primitive:
1. (^ )
Show that:
1. Is the general solution of the differential equation?
2. Is the general solution of the differential equation?
( )
3. Is the general solution of the differential equation?
( )
4. Is the general solution of the differential equation?
( ( ) ) ( )
5. Is the general solution of the differential equation?
6. Is the general solution of the differential equation?
First Order Differential Equations We Can Already Solve
The general solution of ODE involves arbitrary constants. We can be solved directly by integration on both sides.
Example: Find the general solution of the following ODEs:
1.
Solution:
Integrate both sides, we get
Solution:
Integrate both sides, we get
Solution:
Integrate both sides, we get (^ )
Exercises: Solve the following differential equations:
3. 2.^ 1.
Whence we get the general solution:
Example: Solve the following the differential equation:
( ) ( )
Solution:
Separating variable, we get
Or. /. /
Integrating, we get ∫. / ∫. /
Which, is | |^ | |
Or |^ |
Whence we get the general solution:
Exercises: Solve the following differential equations:
1. (^ )^ ; 2. (^ ) 3. (^ )^ ; 4. 5. ( ) 6. (^) √ ( ) 7. 8. (^ ) 9. ( ) ( ) ; 10. (^) √ ( )^ ( ) 11. √ √ 12. (^ )^ √ 13. ( ) (^) √ ( ) **14.
16.** (^ )
Homogeneous First Order Differential Equation
The equation is called homogeneous if P and Q are homogeneous function of x and y of the same order it is reduced to the form
And is solved by the substitution, , we obtain
( ) ( )
Therefore,
And so
( )
This is separable differential equation. Separate the variables and integrate both sides, we get
And then may be replaced with
Note: The function (^ )^ is called homogeneous of the degree if
( ) ( )
Example: Solve the following differential equation:
Solution:
Dividing both the numerator and the denominator by, , we get
Making the substitution
Separating variables, and integrate both sides, we get
Which, is
Or
Substitution, , we get | |
We get the general solution:
. /
Exercises:
Solve the following differential equations:
1. ( ) ( ) 2. 3. (^ ) 4. 5.. /. / 6. √ 7. √ ( ) 8. (^ ) 9. (^ ) 10. (^ ) 11. (^ )^ ; 12. (. / ( ) √
Linear First Order Differential Equation
A differential equation is called linear if it's of the first degree with respect to the required function y and all of its derivatives. A linear equation of the first order has the form
The complete integral of the linear equation will be of the form
∫ (^) {∫ ∫ (^) }
Example: Solve the equation:
Solution:
Comparing by the linear equation, we get
Substituting into the integral form, we get
{∫( ) ∫.^ /^ }
Integrating, we get
( ) (^) {∫( ) ( ) (^) }
Example: Solve the differential equation:
Solution:
Bernoulli's Equation
We consider an equation of the form
Where and are continuous functions of (or constant), and and (otherwise we would have a linear equation).
This equation is called Bernoulli's Equation and reduces to a dividing all terms of the equation by, , we get
Making the substitution
We have ( )
Substituting into the above equation, we get
( ) ( )
This is linear equation.
Finding it’s the complete integral and Substituting for ; we get the complete integral of the Bernoulli's.
Example: Solve the equation:
Solution:
Comparing by Bernoulli's equation, we get
Substituting into the linear equation (^ )^ (^ )^ ,
We get
Where
Let us find its complete integral
Consequently, the complete integral of the given equation:
Exercises:
Solve the following differential equations:
**1.
3.**
4. (^ ) 5. 6. ( ) 7. (^ ) 8. (^ ) 9. (^ )
Example: Solve the differential equation:
Solution:
We have
The condition is fulfilled.
Integrating, we find:
And ∫( ) ∫
∫(^ )
Example: Find the value of the constant such that the differential equation:
( ) (^) ( ) Is exact
Solution:
Since the differential equation is exact, we have
Exercises:
Integrate the following exact differential equations:
3. (^ )
5. (^ )^ (^ )
7. (^ )
9. (^ )
10. (^ )^ (^ )
11. (^ )^ (^ )
Integrating Factor
If in the differential equation
Then for certain conditions there exist a function ( ) such that
This function ( ) is called the integrating factor.
The integrating factor is readily found in the following cases:
- If (^ )^ then ∫^ (^ )
- If ( ) then ∫^ ( )
- If then
- If then
The cases,(3), and (4), if the equation, is homogeneous.
Example: Find the integrating factor the following differential equation:
( )
Solution:
Higher-order Differential Equations
1. An equation of the form (^ )^ ( ) is solved by integration of the right-hand member.
Example: Solve the equation:
Solution:
The equation (^ )^ not containing in an explicit form, by means of the substitution is reduced to the form ( )
Example: Solve the equation:
Solution:
Put we get or (is linear)
3. The equation ( ) not containing in an explicit form, by means of the
substitution is reduced to the form. /
Example: Solve the equation:
Solution:
Put and consider as a function of then we get
Or
Integrating this equation, we find
Or Or √
√ (^) Or √
Or √
Whence
Finally, we get
{ √^ √( ) }
Exercises:
Solve the following differential equations:
**1.
4.**
