MAPLE: Solving Differential Equations

Includes Laplace Transforms

BEFORE TRYING TO SOLVE DIFFERENTIAL EQUATIONS, YOU SHOULD FIRST STUDY
Help Sheet 3: Derivatives & Integrals.

Derivatives of functions. Recall that if f is a known function of x, then
> diff( f, x ) ; gives f '(x)
> diff( f, x$2 ) ; gives f ''(x)
> diff( f, x$3 ) ; gives f ⁽³⁾(x), etc.
Defining an ordinary differential equation, for example

y'' + 4 y' + 13 y = cos 3x
> de := diff(y(x),x$2) + 4*diff(y(x),x) + 13*y(x)
> =cos(3*x);
Note: When defining a differential equation, include the independent variable; for example, enter diff( y(x), x$2 ), not diff( y, x$2 ).
Solving the ordinary differential equation for y(x)
> Y := rhs( dsolve(de, y(x)) );
The solution is called Y.
Solving the ordinary differential equation subject to initial conditions. For example, solve the initial value problem

y'' + 4y' + 13y = cos 3x
y(0) = 1, y'(0) = 0

> de := diff(y(x),x$2) + 4*diff(y(x),x) + 13*y(x)
> = cos(3*x) ;
> Y := rhs( dsolve( { de, y(0) = 1, D(y)(0) = 0 }, y(x) ) ) ;
The solution is called Y.
> plot( Y, x = 0..5 ) ;
plots the solution Y from x = 0 to 5
Another example. Solve the initial value problem

y⁽⁴⁾ + 10y''' + 38y'' + 66y' + 45y = 4
y(0) = 1, y'(0) = 0, y''(0) = -1, y'''(0) = 2

> de := diff(y(x),x$4) + 10*diff(y(x),x$3) +
> 38*diff(y(x),x$2) + 66*diff(y(x),x) +
> 45*y(x) = 4 ;
> Y := rhs( dsolve( { de, y(0) = 1, D(y)(0) = 0,
> D(D(y))(0) = -1, D(D(D(y)))(0) = 2 }, y(x) ) ) ;
The solution is called Y.
> plot( Y, x = 0..5 ) ;
plots the solution Y from x = 0 to 5
Another example. Solve the initial value problem

y'' + w² y = cos x
y(0) = 1, y'(0) = -2

where w is a constant parameter.
> de := diff(y(x),x$2) + w^2*y(x) = cos(x) ;
> Y := rhs( dsolve( { de, y(0) = 1, D(y)(0) = -2 }, y(x) ) ) ;
The solution is called Y.
> plot( Y, x = 0..5 ) ;
produces an error since you did not specify a value for w
> plot( subs( w = 3, Y ), x = 0..5 ) ;
plots the solution Y from x = 0 to 5 with w set to 3
Other maple tools for solving and plotting solutions of differential equations are found in the DEtools package.
> with( DEtools ) :
> ?DEtools for a list of commands in the DEtools package
- Some examples:
  > ?DEplot
  > ?DEplot1
  > ?DEplot2
  > ?phaseportrait
  > ?dfieldplot
Of course, not every conceivable differential equation can be solved, which is why we still need to know Numerical Methods!
Laplace Transforms. To determine the Laplace transform of a function, say

f(t) = cos t
> with( inttrans ) : load the integral transform package
> f := cos(t) ; defines f as an expression
> F := laplace( f, t, s ) ; stores the Laplace transform of f in F
> F := s/(s^2-25) ; defines F as an expression
> f := invlaplace( F, s, t ) ; stores the inverse Laplace transform of F in f
> G := s/(s^2-9) ; defines G as an expression
> g := invlaplace( G, s, t ) ; stores the inverse Laplace transform of G in g
> f := Heaviside(t-4) ; defines f as the unit step function about t=4
Note: The unit step function is not U(t-4).
> F := laplace( f, t, s ) ; stores the Laplace transform of f in F
> f := t^2 * Heaviside(t-4) ;
> F := laplace( f, t, s ) ; stores the Laplace transform of f in F
> ?inttrans for a list of commands in the inttrans package

Written and Maintained by

Prof. Kevin G. TeBeest
Applied Mathematics
Kettering University
Last modified: 07/23/07

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