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Differential Equations

These equations, containing a derivative, involve rates of change – so often appear in anengineering 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 highestderivative used.

  Examples :-

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Types of differential equations :-

First order Differential EquationsFirst order Linear  Differential EquationsSecond order  Linear Differential EquationsSecond order non – homogeneous Differential Equations

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Examples of Differential Equations

 

 

First order Differential Equations

Solving by direct integration

The general solution of differential equations of the form

 can be found using direct integration.

Substituting the values of the initial conditions will give

Example

Solve the equation

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Example

 Find the particular solution of the differential equation

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   given  y = 5 when x = 3

 

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Example

 A straight line with gradient 2 passes through the point  (1,3). Find the equation of the line.

  

 

 A variables separable differential equation is one in which the equation can be written with all the terms for one variable on one side of the equation, and the other terms on the other side.

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Example

  Find the general solution of the differential equation

           

  

Example

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 Find the general solution of the differential equation

       

 

Example

 Find the particular solution of the differential equation

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  given y = 2 when x = 1

  

Partial fractions are required to break the left hand side of the equation into a form whichcan be integrated.

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so

which integrates to general solution

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substitute values for particular solution

 

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Linear  Differential Equations

 These are first degree differential equations.

   describes a  general linear differential equation  of order n, where an(x), an-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

 

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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

3) Write 

 

4) Integrate the right hand side,

use integration by parts if necessary

 

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5) Divide both sides by the integrating factor.This gives the General solution.

6) Use any initial conditions to find particular solutions.

 

 Example

Find a general solution of the equation

    

    

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so

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Example

Find a general solution of the equation

 

   where x ≠2 , and hence find the particular solution for y = 1 when x=-1

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Second order  Linear Differential Equations

To solve equations of the form

1) Write down the auxiliary equation am2 +bm + c = 0         

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      (why this works, UCL.ac.uk)

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

6) Use any initial conditions to find the particular solution.

 

Example

 Find the general solution of the equation

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  and the particular solution for which y = 7 when x=0 and dy/dx = 7

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Example

  Find the general solution of the equation

  

  and the particular solution for y=0 and dy/dx = 3 when x=0

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Example

 Find the general solution of the equation

 

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Second order non – homogeneous Differential Equations

      The solution to equations of the form

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          has two parts, the complementary function (CF) and the particular integral (PI).

so 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 +DIf Q(x) is quadratic, try Cx2 +Dx +EIf Q(x) is wave function, try CSinx +DcosxIf Q(x) is a constant, try y = CIf Q(x) is ekx, try y = Cekx

The PI cannot have the same form as any of the terms in the CF, so care has to be taken to ensure that this is not the case.

In such a situation, an extra x term is usually introduced to the PI.

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  A particular solution is found by substituting initial conditions into the general solution. Donot just use the CF!!!

Example

Find the general solution of the equation

     

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Example

  Find the general solution of the equation

    

  and the particular solution for y=0 and dy/dx = 5 when x=0

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Now, substitute these back into the original equation

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Now find the particular solution

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Phew!!

 

More Info

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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

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