Your Federal Quarterly Tax Payments are due April 15th Get Help Now >>

First Order Differntial Equation by ugl97184

VIEWS: 18 PAGES: 22

First Order Differntial Equation document sample

More Info
									                           PROGRAMME 24



                              FIRST-ORDER
                              DIFFERENTIAL
                               EQUATIONS

STROUD   Worked examples and exercises are in the text
               Programme 24: First-order differential equations

Introduction
Formation of differential equations
Solution of differential equations
Bernoulli’s equation




STROUD     Worked examples and exercises are in the text
               Programme 24: First-order differential equations

Introduction
Formation of differential equations
Solution of differential equations
Bernoulli’s equation




STROUD     Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Introduction



 A differential equation is a relationship between an independent variable x, a
 dependent variable y and one or more derivatives of y with respect to x.

 The order of a differential equation is given by the highest derivative involved.

                      dy
                  x       y 2  0 is an equation of the 1st order
                      dx
            d2y
          xy 2  y 2 sin x  0 is an equation of the 2nd order
             dx
          d3y     dy
             3
               y     e 4 x  0 is an equation of the 3rd order
          dx      dx



STROUD        Worked examples and exercises are in the text
               Programme 24: First-order differential equations

Introduction
Formation of differential equations
Solution of differential equations
Bernoulli’s equation




STROUD     Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Formation of differential equations



 Differential equations may be formed from a consideration of the physical
 problems to which they refer. Mathematically, they can occur when
 arbitrary constants are eliminated from a given function. For example, let:

                                         dy
      y  A sin x  B cos x so that          A cos x  B sin x therefore
                                         dx
      d2y
         2
             A sin x  B cos x   y
      dx

                                 d2y
      That is                          y0
                                 dx 2


STROUD          Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Formation of differential equations



 Here the given function had two arbitrary constants:

                            y  Asin x  B cos x

 and the end result was a second order differential equation:

                                d2y
                                   2
                                      y0
                                dx

 In general an nth order differential equation will result from consideration
 of a function with n arbitrary constants.



STROUD        Worked examples and exercises are in the text
               Programme 24: First-order differential equations

Introduction
Formation of differential equations
Solution of differential equations
Bernoulli’s equation




STROUD     Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Solution of differential equations
Introduction
Direct integration
Separating the variables
Homogeneous equations – by substituting y = vx
Linear equations – use of integrating factor




STROUD         Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Solution of differential equations
Introduction


 Solving a differential equation is the reverse process to the one just
 considered. To solve a differential equation a function has to be found for
 which the equation holds true.

 The solution will contain a number of arbitrary constants – the number
 equalling the order of the differential equation.

 In this Programme, first-order differential equations are considered.




STROUD         Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Solution of differential equations
Direct integration


 If the differential equation to be solved can be arranged in the form:

                               dy
                                   f ( x)
                               dx

 the solution can be found by direct integration. That is:


                             y   f ( x)dx




STROUD        Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Solution of differential equations
Direct integration


 For example:
                           dy
                               3x 2  6 x  5
                           dx
 so that:
                        y   (3 x 2  6 x  5)dx
                           x3  3x 2  5 x  C

 This is the general solution (or primitive) of the differential equation. If a
 value of y is given for a specific value of x then a value for C can be found.
 This would then be a particular solution of the differential equation.



STROUD        Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Solution of differential equations
Separating the variables


 If a differential equation is of the form:

                                dy f ( x)
                                  
                                dx F ( y )

 Then, after some manipulation, the solution can be found by direct
 integration.

                 F ( y )dy  f ( x)dx so    F ( y)dy   f ( x)dx



STROUD        Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Solution of differential equations
Separating the variables


 For example:
                               dy   2x
                                  
                               dx y  1
 so that:
                ( y  1)dy  2 xdx so    ( y  1)dy   2 xdx
 That is:
                           y 2  y  C1  x 2  C2
 Finally:
                             y 2  y  x2  C



STROUD      Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Solution of differential equations
Homogeneous equations – by substituting y = vx


 In a homogeneous differential equation the total degree in x and y for the
 terms involved is the same.
 For example, in the differential equation:
                                dy x  3 y
                                   
                                dx   2x

 the terms in x and y are both of degree 1.

 To solve this equation requires a change of variable using the equation:
                                  y  v( x ) x



STROUD        Worked examples and exercises are in the text
                 Programme 24: First-order differential equations

Solution of differential equations
Homogeneous equations – by substituting y = vx


 To solve:
                           dy x  3 y
                              
                           dx   2x
 let
                             y  v( x ) x
 to yield:
                  dy       dv     x  3 y 1  3v
                     v x    and        
                  dx       dx       2x       2
 That is:
                                 dv 1  v
                             x      
                                 dx   2
 which can now be solved using the separation of variables method.


STROUD       Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Solution of differential equations
Linear equations – use of integrating factor


 Consider the equation:
                                     dy
                                         5 y  e2 x
                                     dx
 Multiply both sides by e5x to give:

                e5 x
                       dy
                       dx
                           e5 x 5 y  e5 x e2 x that is
                                                         d
                                                         dx
                                                             ye5 x   e7 x
 then:
                 d  ye    e          dx so that ye5 x  e7 x  C
                           5x        7x



 That is:
                                   y  e 2 x  Ce 5 x


STROUD        Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Solution of differential equations
Linear equations – use of integrating factor


 The multiplicative factor e5x that permits the equation to be solved is
 called the integrating factor and the method of solution applies to
 equations of the form:


                 Py  Q where e 
             dy                    Pdx
                                       is the integrating factor
             dx

 The solution is then given as:


                      y.IF   Q.IFdx where IF  e 
                                                       Pdx




STROUD        Worked examples and exercises are in the text
               Programme 24: First-order differential equations

Introduction
Formation of differential equations
Solution of differential equations
Bernoulli’s equation




STROUD     Worked examples and exercises are in the text
                   Programme 24: First-order differential equations

Bernoulli’s equation



 A Bernoulli equation is a differential equation of the form:

                                dy
                                    Py  Qy n
                                dx
 This is solved by:

 (a) Divide both sides by yn to give:
                                   dy
                             yn       Py1 n  Q
                                   dx
 (b) Let z = y1−n so that:
                             dz                 dy
                                 (1  n) y  n
                             dx                 dx

STROUD        Worked examples and exercises are in the text
                  Programme 24: First-order differential equations

Bernoulli’s equation



 Substitution yields:
                           dz                 dy
                               (1  n) y  n
                           dx                 dx
 then:
                                   dy          
                    (1  n)  y  n     Py1 n   (1  n)Q
                                   dx          
 becomes:
                               dz
                                   P z  Q1
                                     1
                               dx
 Which can be solved using the integrating factor method.


STROUD        Worked examples and exercises are in the text
                         Programme 24: First-order differential equations
Learning outcomes


Recognize the order of a differential equation

Appreciate that a differential equation of order n can be derived from a function
containing n arbitrary constants

Solve certain first-order differential equations by direct integration

Solve certain first-order differential equations by separating the variables

Solve certain first-order homogeneous differential equations by an appropriate
substitution

Solve certain first-order differential equations by using an integrating factor

Solve Bernoulli’s equation.



     STROUD         Worked examples and exercises are in the text

								
To top