Differential Equations

 

These equations, containing a derivative,

involve rates of change – so often appear

in an engineering or scientific context.

Solving the equation involves integration.

 

 

·       The order of a differential equation is given

   by the  highest derivative used.

 

·       The degree of a differential equation is given

 by the  degree of the  power of the highest

 derivative used.

 

 

Examples :-

 

   

 

Types of differential equations :-

 

 

 

 

 

First order Differential Equations

 

Solving by direct integration

 

 

 e.g.

 

 

 

 

v  Example

 

 Find the particular solution of the

 differential equation

 

       given  y = 5 when x = 3

 

 

 

 

v  Example

 A straight line with gradient 2 passes through

 the point  (1,3). Find the equation of the line.

 

  

 

 

 

v  Example

 

 Find the general solution of the differential equation

 

           

 

  

 

v  Example

 

 Find the general solution of the differential equation

 

      

 

 

v  Example

 

 Find the particular solution of the differential equation

 

       given y = 2 when x = 1

 

  

 

 

 

   

 

 

 

Linear  Differential Equations

 

 These are first degree differential equations.

 

 

 

 

 describes a  general linear differential equation  of order n,

 where a_n(x), a_n-1(x),etc and f(x) are given functions

 of x or constants.

 

 Louis Arbogast introduced the differential operator

 D = d/dx  , which simplifies the general equation to

 

 

 

 or

 

 

                                           

 

  If f(x) = 0 , the equation  is called homogeneous.

  If f(x) ≠0 , the equation is non-homogeneous

 

 

 

First order Linear  Differential Equations

 

To solve equations of the form

    

 

1)              express in standard form

 

                 

 

  where  P and Q are functions of x or constants

 

2)            Multiply both sides by the Integrating Factor

 

Write 

 

3)            Integrate the right hand side,

use integration by parts if necessary.

 

 

4)            Divide both sides by the integrating factor.

This gives the General solution

 

5)            Use any initial conditions to find

particular solutions.

 

 

 

 Example

 

  Find a general solution of the equation

 

    

 

    

 

  

 

 

 

Example

 

   Find a general solution of the equation

 

 

 

   where x ≠2 , and hence find the particular solution

  for y = 1 when x=-1

 

 

 

   

 

    

 

 

 

 

 

Second order  Linear Differential Equations

 

      To solve equations of the form

      

 

1.    Write down the auxiliary equation

am² +bm + c = 0

 

2.  Examine the discriminant of

the auxiliary equation.

 

3.  For real and distinct roots,

   m₁ and m₂, the general solution is

 

  

      

4.  For real and equal roots,

   the general solution is

 

 

 

 

5.  For complex conjugate roots,

   m₁= p + iq  and m₂ = p- iq  ,

  the general solution is

 

  

 

 

Use any initial conditions to find the

particular solution.

 

·       Example

 

  Find the general solution of the equation

 

 

 

  and the particular solution for which

  y = 7 when x=0 and dy/dx = 7

   

  

 

 

  

 

·       Example

 

  Find the general solution of the equation

 

  

 

  and the particular solution for

  y=0 and dy/dx = 3 when x=0

 

 

 

   

 

 

    

 

 

 

 

 

 

 

·       Example

 

 Find the general solution of the equation

 

 

 

   

 

 

 

 

Second order non – homogeneous Differential Equations

 

      The solution to equations of the form

 

     

 

          has two parts, the complementary function (CF)

and the particular integral (PI).

 

Q(x) = CF +PI

 

The CF is the general solution as described above

 for solving homogeneous equations .

 

           The Particular Integral is found by substituting

    a form similar to Q(x) into the left hand side equation,

and equating co-efficients.

 

If Q(x) is a linear function, try y = Cx +D

If Q(x) is quadratic, try Cx² +Dx +E

If Q(x) is wave function, try CSinx +Dcosx

If Q(x) is a constant, try y = C

If Q(x) is e^kx, try y = C e^kx

 

  

 

    

 

 

 

 

 

  A particular solution is found by substituting

  initial conditions into the general solution.

 

Do not just use the CF!!!

 

·       Example

 

     Find the general solution of the equation

 

    

 

 

 

   

 

 

       

 

Example

 

  Find the general solution of the equation

 

    

 

  and the particular solution for

  y=0 and dy/dx = 5 when x=0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

More Info

 

http://en.wikipedia.org/wiki/Differential_equation

http://en.wikipedia.org/wiki/Ordinary_differential_equation

http://en.wikipedia.org/wiki/Linear_differential_equation

http://en.wikipedia.org/wiki/Superposition_principle

http://en.wikipedia.org/wiki/Integrating_factor

 

2.  Some examples of differential equations

 

http://en.wikipedia.org/wiki/Examples_of_differential_equations

http://en.wikipedia.org/wiki/RC_circuit

http://en.wikipedia.org/wiki/Classical_mechanics

http://en.wikipedia.org/wiki/Dynamical_systems

http://en.wikipedia.org/wiki/Numerical_methods

http://en.wikipedia.org/wiki/Newton%27s_Laws

http://en.wikipedia.org/wiki/Radioactive_decay

http://en.wikipedia.org/wiki/Wave_equation

http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

http://en.wikipedia.org/wiki/Shallow_water_equations

http://en.wikipedia.org/wiki/Maxwell%27s_equations

http://en.wikipedia.org/wiki/Harmonic_oscillator

http://en.wikipedia.org/wiki/Vector_space

http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients

http://en.wikipedia.org/wiki/Euler%27s_formula

http://en.wikipedia.org/wiki/Poisson%27s_equation

http://en.wikipedia.org/wiki/Quantum_mechanics

http://en.wikipedia.org/wiki/Verhulst_equation