1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

6.2 Partial Fractions and Convolutions 375

2. Solve the initial value problem

u′′+ 2 au′+u= 0 , u( 0 )=u 0 , u′( 0 )=u 1

in these three cases: 0<a<1,a=1,a>1.

3. Solve these nonhomogeneous problems with zero initial conditions.

a.u′+au=1;

c. u′′+ 4 u=sin(t);

e.u′′+ 2 u′= 1 −e−t;

b. u′′+u=t;

d. u′′+ 4 u=sin( 2 t);

f. u′′−u=1.

4. Complete the square in the denominator and use the shift theorem
   [F(s−a)=L(eatf(t))]toinvert

U(s)=

su 0 +(u 1 + 2 au 0 )

s^2 + 2 as+ω^2.

There are three cases, corresponding to

ω^2 −a^2 > 0 , = 0 ,< 0.

5. Use partial fractions to invert the following transforms.

a.^1

s^2 − 4

;

c. s((ss 2 ++^32 ));

b.^1

s^2 + 4

;

d.^4

s(s+ 1 )

.

6. Prove properties (4a) and (4c) of the convolution.


7. Compute the convolutionf∗gfor

a.f(t)=1,g(t)=sin(t);

b.f(t)=et,g(t)=cos(ωt);

c. f(t)=t,g(t)=sin(t).

8. Demonstrate the following properties of convolution either directly or by
   using Laplace transform.
   a. 1 ∗f′(t)=f(t)−f( 0 );
   b.(t∗f(t))′′=f(t);
   c. (f∗g)′=f′∗g=f∗g′,iff( 0 )=g( 0 )=0.