12.8 • MULTIPLICATION AND D IVISION BY t 561- EXAMPLE 12.23 Use Laplace transforms to solve the initial value problem
ty" (t) -ty' (t) - y (t) = 0, with y (O) = 0.
Solution If we let Y ( s) denote the Laplace transform of y ( t) and substitute
Equations (12-32) and (12-33) into the preceding equation, we get
- s^2 Y' (s) -2sY (s) + 0 + sY' (s) + Y (s) -Y (s) = 0. (12-34)
Equation (12-34) involves Y' (s) and can be written as a first-order linear differ-
ential equat ion
2
Y' (s) + s _
1
Y (s) = O.
The integrating factor p for the differential equation is
p=exp(j s~ Ids) =e2 1 n(s- 1) = (s - 1)2.
Multiplying Equation (12-35) by p produces(s - 1)^2 Y' (s) + 2 (s - 1) Y (s) =! [ (s - 1)2 Y (s)] = O.
(12- 35 )When we integrate the equation :s [(s - 1)^2 Y(s)] = 0 with respect to s, the
result is (s - 1)
2
Y (s) = C , where C is the constant of integration. Hence the
solution to Equation (12-34) isc
Y(s)= 2.
(s - 1)The inverse of the transform Y (s) in this equation is the desired solution:-------... EXERCISES FOR SECTION 12.8- Find .C (te-^2 t).
- Find .C (t^2 e^4 ').
- Find .C (tsin3t).
- Find .C (t^2 cos2t).
- Find .C (t sinh t).