248 Ordinary Differential Equations
the proper syntax requesting a CAS to use the Laplace transform method to
solve the initial value problem y" + 4y = 0; y(O) = 1, y'(O) = -1, then the
1
CAS will simply respond y( x ) = -
2
sin 2x + cos 2x.EXERCISES 5.2
In exercises 1-7 use the Laplace transform method to calculate
the general solution of the given differential equation manually. If
you have a CAS available which contains algorithms for using the
Laplace transform method, use them to calculate the solutions of
exercises 1-7 and compare those answers to the ones you obtained
by hand.
- y'-y=O
- y" - 2y' + 5y = 0
- y' + 2y = 4
- y" - 9y = 2 sin 3x
- y" + 9y = 2 si n 3x
- y" + y' - 2y = xex - 3x^2
- y(4) - 2y(3) + y(2) = xex - 3x 2
In exercises 8-14 use the Laplace transform method to calculate
the solution of the given initial value problem manually. If you have
a CAS available which contains algorithms for using the Laplace
transform method to solve initial value problems, use them to cal-
culate the solutions of exercises 8-14 and compare those answers to
the ones you obtained by hand.
8. y' =ex; y(O) = - 1
9. y' - y = 2ex; y(O) = 1
10.11.y" - 9y = x + 2; y(O) = -1,
y" + 9y = x + 2; y(O) = -1,
y'(O) = 1y'(O) = 1- y"-y'+6y=-2sin3x; y(O)=O, y'(O)=-l
13. y" - 2y' + 2y = 1-x^2 ; y(O) = 1, y'(O) = 0
- y"' + 3y" + 2y' = x + cosx; y(O) = 1, y'(O) = -1, y"(O) = 2