1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans! orm Method 273

EXERCISES 5.5

Find the solution of the following initial value problems using the

Laplace transform method.

l. y' + 3y = o(x - 2); y(O) = 0

2. y' -3y = o(x - 1) + 2u(x -2); y(O) = 0


3. y" + 9y = o(x -n) + o(x -3n); y(O) = 0, y'(O) = 0


4. y" -2y' + y = 2o(x -1); y(O) = o, y'(O) = 1


5. y" -2y' + 5y = cosx + o(x - n); y(O) = 1 , y'(O) = 0


6. y" + 4y = o(x -n) cosx; y(O) = 0, y'(O) = 1


7. y" + a^2 y = o(x - n)f(x); y(O) = 0, y'(O) = 0
   where a is a real constant and f(x) is a function that is continuous on
   some interval about the point x = n.

I Comments on Computer Software I The so lution of the initial value problem

y" + 2y' + 5y = 50(x -2); y(O) = 4, y' (0) = 0

of example 2 can be computed using the following two MAPLE statements.

DE:=diff(y(x), x$2) + 2diff(y(x), x) + 5 y(x) = 5*Dirac(x - 2);

dsolve( {DE, y(O) = 4, D(y)(O) = O}, y(x), method=laplace);

The first statement specifies the differential equation to be solved. Observe
that MAPLE uses the notation "Dirac(x - 2)" to represent the Dirac delta
function o(x - 2). The second statement instructs the computer to use the
Laplace transform method to solve the differential equation using the specified
initial condition values and to print the solution. The printed solution contains
the Heaviside function and is equivalent to the solution of example 2.