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Differential Equations

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Ordinary Differential Equation

An #~{ordinary differential equation} is one which involves derivatives of a function, _ ~y , _ of one variable _ ~x :

~F rndb{{fract{d^~n ~y,d~x^~n}, ... ,fract{d ~y,d~x}, ~y, ~x }} _ = _ 0

[This is in contrast to #~{partial differential equations} which are equations involving partial derivatives of functions of more than one variable.]

Order of Differential Equation

The #~{order} of an ordinary differential equation is the order of the highest derivative in the equation.

For example:

script{rndb{fract{d^2~y,d~x^2}},,,3,} + 2 script{rndb{fract{d~y,d~x}},,,5,} + 5 ~y^2 - 30 ~x _ = _ 20

is a second order equation.

Degree of Differential Equation

The #~{degree} of an ordinary differential equation is the degree (power) of the derivative of highest order.

So

script{rndb{fract{d^2~y,d~x^2}},,,3,} + 2 script{rndb{fract{d~y,d~x}},,,5,} + 5 ~y^2 - 30 ~x _ = _ 20

is of degree 3.

 

Linear Differential Equation

An ordinary differential equation is #~{linear} if all the derivatives have degree one

~a_~n fract{d^~n~y,d~x^~n} + ... + ~a_1 fract{d~y,d~x} + ~a_0 ~y _ = _ ~f ( ~x )

where ~a_~n, ... ~a_0 = are constants, and ~f ( ~x ) is a function involving only terms with ~x or constants. The above is therefore an ~n^~{th} order linear differential equation (#~{l.d.e.}).

Arbitrary Constants

Differential equations arise in many areas of mathematics and physics, resulting from a set of conditions, such as the laws of motion in kinematics. In general a solution is required for the differenetial equation which will then be used to elucidate some aspect of the problem being studied

As in solving other equations, such as polynomials, the solution should be "fed back" into the equation, to see if it really does agree. The question is, if we do find #a solution, is this enough?

Let us take a simple example, a second order linear equation:

fract{d^2~y,d~x^2} + ~y _ = _ 0

and lets try the 'solution' _ ~y = sin ~x . _ Well , _ d^2 ( sin ~x ) ./ d~x^2 = - sin ~x , _ so this is a solution of the equation. However you may notice that if we multiply the 'solution' by any constant, ~A, say, then _ d^2 ( ~A sin ~x ) ./ d~x^2 = - ~A sin ~x , _ so _ ~y = ~A sin ~x _ is also a solution of the equation.

But _ d^2 ( cos ~x ) ./ d~x^2 = - cos ~x , _ so for any ~B , _ ~y = ~B cos ~x _ is also a solution, and as differentiation is a linear operator,

_ _ _ _ ~y _ = _ ~B cos ~x + ~A sin ~x

is a solution for any pair of arbitrary constants ~A and ~B.

In the same way as when we integrate a function we get one arbitrary constant, then the general solution a differential equation of degree ~n will have ~n arbitrary constants. Note, however, that finding a solution with ~n constants does not mean that this is necessarily the most general solution. This can be seen with the current example when we deal with second order homogenous equations .]

We can take the solution of our example a bit further, because we can express it as

_ _ _ _ ~y _ = _ ~C cos &phi. cos ~x + ~C sin &phi. sin ~x _ = _ ~C cos ( ~x - &phi. )

where _ ~C = &sqrt.${~A&powtwo. + ~B&powtwo.} , _ &phi. = cos^{-1} ~B ./ ~C . _ So now these are our two arbitrary constants. Similarly

_ _ _ _ ~y _ = _ ~C sin &psi. cos ~x + ~C cos &psi. sin ~x _ = _ ~C sin ( ~x + &phi. )

where _ ~C = &sqrt.${~A&powtwo. + ~B&powtwo.} , _ &psi. = sin^{-1} ~B ./ ~C .

This demonstrates that there may be several ways of presenting the "general" solution of a differential equation, so care must be taking when checking answers to exercises.

Initial Conditions

As we have seen, the general solution to a ~n^{th} order differential equation will involve ~n arbitrary constants. In most actual problems, however, there will be a set of conditions, called #~{initial conditions}, which will determine some or all of these constants.

As a very simple example, consider the equation

fract{d~y,d~x} _ = _ ~x

subject to the initial condition _ ~y = 0 _ when _ ~x = 0.

The general solution is obtained by simple integration, _ ~y = ~x^2 ./ 2 + ~c .

By the initial condition we have: _ 0 = 0 + ~c _ => _ ~c = 0 .

The solution of the equation using the initial condition(s) is called the #~{particular solution}, which in this case is _ ~y = ~x^2 ./ 2 .