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Solving Differential Equations

Includes Laplace Transforms

Before Trying To Solve Differential Equations, You Should First Study Help Sheet 3: Derivatives & Integrals.

  1. 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.

  1. 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 ), notdiff( y, x$2 ).

  1. Solving the ordinary differential equation for y(x)

    > Y := rhs( dsolve(de, y(x)) ); 

The solution is called Y.

  1. 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

  1. 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

  1. 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

  1. 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!

  1. Laplace Transforms. To determine theLaplace 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