ADVANCED MATHEMATICS FORM 6 – DIFFERENTIAL EQUATION

Share this post on:

A differential equation is a relationship between the rates of different variables (i.e an independent variables x, a dependent variable, y, and one or more differential coefficient of y with respect to x).

          Eg        edu.uptymez.com + y sin x = 0

                    xyedu.uptymez.com + y edu.uptymez.com + edu.uptymez.com = 0

                      edu.uptymez.com + edu.uptymez.com – 4y = 0

The order of a differential equation

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

          edu.uptymez.com – y2 = 0 is an eqn of the 1st order

          xyedu.uptymez.com – y2 sin x = 0 is an eqn of the 2nd order

          xyedu.uptymez.com – yedu.uptymez.com +edu.uptymez.com = 0 is an eqn of the 3rd order

The degree of a differential equation

          Eg edu.uptymez.com + y sin x = 0 the degree is 1b

Linear differential equation (L.D.E)

A linear differential equation should look in the form

          edu.uptymez.com + edu.uptymez.com + edu.uptymez.com + ……… edu.uptymez.com edu.uptymez.com + pn y = Q

          Where P1, P2, P3 …Pn, Q are functions of x, or constant n = 1, 2, 3, 4…….

               Examples

          i) edu.uptymez.com + edu.uptymez.com + 6y = 0

          ii) edu.uptymez.com + edu.uptymez.com + edu.uptymez.com = edu.uptymez.com

          iii) edu.uptymez.com  = 4

Examples of non L.D.E

edu.uptymez.com + edu.uptymez.com + 6y = 0

 edu.uptymez.com + edu.uptymez.com + 6y = 0

Note

A D. E will be linear if

  •          Variable y and its derivatives occur at the first degree only
  •          No product of y and its derivatives
  •         No transcendental function of y or x

 

edu.uptymez.com

Exercise

Show which of the following differential equations is the L.D.E or N.L.D.E

i) edu.uptymez.com = y……………linear

ii) (x2 – 1) edu.uptymez.com = y……linear

iii) edu.uptymez.com + y2 + 4 = 0…………..non – linear

iv) edu.uptymez.com = edu.uptymez.com……………non – linear

v) edu.uptymez.com = 1 + xy………linear

vi) (x2 – y2) edu.uptymez.com = 2xy……Non – linear

Formulation of a differential

Differential equation may be formed when arbitrary constant are eliminated from a given function.

Examples 1

     Y = A sin x + B cos x, where A and B are two arbitrary constants

     Solution

        edu.uptymez.com = A cos x – B sin x

         edu.uptymez.com = -A sin x – B cos x

 edu.uptymez.comhis is identical to the original eqn with opposite signs

          edu.uptymez.com = -1(A cos x + B sin x)

          edu.uptymez.com = -y

Form a DE whose solution is the form y = Ae2x + Be-3X  Where A and B are constant.

Example 2

Form a different equation from the function y = x + edu.uptymez.com

          Solution
y = x + edu.uptymez.com = x + Ax-1

                   edu.uptymez.com = 1 – Ax-2 = 1 – edu.uptymez.com

          From the given equation

                   Y = x + edu.uptymez.com

                   yx = x + edu.uptymez.com

                   x(y – x) = A

                   edu.uptymez.com = 1- x (y – x) x-2

                       = 1 – edu.uptymez.com

                   edu.uptymez.com = x2 – x (y – x)

                   edu.uptymez.com = x – (y – x)

                                edu.uptymez.com = 2x – y

Example 3

Form the DE for y = Ax2 + Bx

                   Y = Ax2 + Bx…… (i)

                   edu.uptymez.com = 2Ax + B

                   edu.uptymez.com – 2Ax = B……. (ii)

                   edu.uptymez.com = 2A

              edu.uptymez.com

                   Substituting both (ii) and (iii) into (i) gives
edu.uptymez.com

                          edu.uptymez.com

                             edu.uptymez.com – 2xedu.uptymez.com + 2y = 0

          Solution to a DE

This involves finding the function for which the equation is true (i.e manipulating the eqn so as to eliminate all the differential coefficients and have a relationship between x and y)

E.g. verify that the function

i) Y = 3e2x is a soln of DE = edu.uptymez.com – 2y = 0 for all x

ii) y (x) = sin x – cos x + 1 is a soln of eqn edu.uptymez.com + y = 1 for all value of x

           Solution
edu.uptymez.com

 Substitute equation iv into iii.

         edu.uptymez.com= 2y
edu.uptymez.com – 2y = 0

  Direct integration
Direct integration is used to solve equation which is arranged in the form edu.uptymez.com = f (x)

Examples

1.  Solves  edu.uptymez.com = 3x2 – 6x + 5

          Solution

                   edu.uptymez.com = 3x2 – 6x + 5

          Then edu.uptymez.com = edu.uptymez.com

                             = x3 – 3x2 + 5x + c

                             Y = x3 – 3x2 + 5x + c

2.  Solve edu.uptymez.com = 5x3 + 4

          Solution

                Rearranging in the form edu.uptymez.com = f (x)

                edu.uptymez.com = 5x2 + edu.uptymez.com

               Then  edu.uptymez.com = edu.uptymez.com dx

                             Y = edu.uptymez.com + 4ln x + c

                             Y = edu.uptymez.com + 4ln x + c

          The solution is a called general solution since it consist a constant C (i.e. unknown constant)

3. Find the solution of the eqn ex
edu.uptymez.com = 4 given that y = 3

          When x = 0

          Soln: rearrange the given equation

                   = edu.uptymez.com = 4e-x + 7

          This is called a particular soln since it contains a non – unknown       variable.

First order D.E

Separating the variables

 This is a method used to solve the D.E when is in the form edu.uptymez.com = f (x, y)

  The variables y on the right hand-side prevents solving by direct integration

Example

1. Solve  edu.uptymez.com = edu.uptymez.com

          Solution  edu.uptymez.com = edu.uptymez.com

                   = (y + 1) edu.uptymez.com = 2x

          Integrating both sides with respect to x

                 edu.uptymez.com

2. Solve  edu.uptymez.com =  edu.uptymez.com

                   Solution

edu.uptymez.com =  edu.uptymez.com

                     edu.uptymez.com  edu.uptymez.com =  edu.uptymez.com

                   edu.uptymez.com dy = edu.uptymez.com dx

            edu.uptymez.com

                   edu.uptymez.com

 3.   By separating variables solve the differential equation (xy + x) dx =(x2y2 + y2 + x2 + 1) dy

          Solution

          Given Equation;

                   (xy + x) dx = (x2y2 + y2 + x2 + 1) dy

                   X (y + 1) dx = (y2 (x2 + 1) + (x2 +1)) dy

                   X (y + 1) dx = (x2 + 1) (y2 + 1)  dy

                             edu.uptymez.com dx =  edu.uptymez.com dy

          Integrating both sides

                   edu.uptymez.com dx =  edu.uptymez.com dy

R.H.S =edu.uptymez.com= y-1+edu.uptymez.com  

edu.uptymez.com dx =edu.uptymez.com) dy

                   ½ ln (x2 + 1) = edu.uptymez.com – y + 2 ln (y + 1) + C

Share this post on: