Additional Exercises Handout, Math 344 Al Jimenez, Cal Poly 1/20/2012

Additional Exercises Handout for Math 344

For Section 8.3 Periodic Functions and Laplace Transform

1. Calculate the solution y(t) for the IVP ( ), (0) (0) 0y y f t y y′′ ′+ = = = with period π, and one period

described using the Heaviside function by [ ]( ) sin ( ) ( )pf t t t t π= − −U U (this is a full-wave rectifier input.)

(NOTE: The inverse Laplace:

( )1

22

1 1(sin cos )

21t t t

s

− = − +

L

For Section 9.6 Bessel’s Equation

Use Bessel’s parametric equation and its general solution to write the solution of the following second order

differential equations:

1. 2 2 1( ) 0

3x y xy x y′′ ′+ + − =

2. 2 2( 1) 0x y xy x y′′ ′+ + − =

3. 2 24 4 (4 25) 0x y xy x y′′ ′+ + − =

4. 2 216 16 (16 1) 0x y xy x y′′ ′+ + − =

5. 0xy y xy′′ ′+ + =

6. [ ] 40

dxy x y

dx x

′ + − =

7. 2 2(9 4) 0x y xy x y′′ ′+ + − =

8. 2 2(36 2) 0x y xy x y′′ ′+ + − =

Answers to odd problems:

1111. Since 1/ 3p = the general solution is 1 21/ 3 1/ 3

( ) ( )y c J x c J x−

= + .

3. Since p = 5/2 the general solution is 1 5/ 2 2 5/ 2( ) ( )y c J x c J x−= + .

5. Since p = 0 the general solution is y = c1J0(x) + c2Y0(x). 7. Since p = 2 and α = 3 the general solution is y = c1J2(3x) + c2Y2(3x).

For Section 4.11 Inner Product Spaces 1. Calculate the projection or Best Approximation, of sin x onto x , with the inner product

( ), ( ) ( ) ( )f x g x f x g x dxπ

π−= ∫ . An interpretation of this result is that the best we can approximate sin x

given only the function x is 2

3x

π on the interval [–π, π] with a least square error measure.

Additional Exercises Handout, Math 344

2. Calculate the projection or Best Approximation, of sin

2

0( ), ( ) ( ) ( )f x g x f x g x dx

π

= ∫

3. Show that 2

3c

π= is the solution to the Least Square Minimization problem

( )2 2min ( ) sin sinc

c x cx dx x cxπ

πε

−= − = −∫

Approximation in exercise 1 above

For Section 4.12 Orthogonal Vectors and Gram

1. Calculate the Best Approximation of

at right using the inner product

orthogonal set { }1 2 3, , 1, ,P P P x x = −

(Answer is: 21

(13 15 )16

x− )

2. Show that the Chebyshev polynomials

1

21

( ) ( ),

1

g x h xg h dx

x−=

−∫ . NOTE

(12 2 1 22

1(1 ) sin ( ) 1

2x x dx x x x

− −− = − −∫

For Section 10.6

, Math 344 Al Jimenez, Cal Poly

Calculate the projection or Best Approximation, of sin x onto cos x using

( ), ( ) ( ) ( )f x g x f x g x dx

is the solution to the Least Square Minimization problem

2 2min ( ) sin sinc x cx dx x cx= − = − (the same as obtained by using projections or Best

in exercise 1 above).

Orthogonal Vectors and Gram-Schmidt

Calculate the Best Approximation of f(x) shown on the sketch 1

1, ( ) ( )g h g x h x dx

−= ∫ onto the

2 1, , 1, ,

3P P P x x

= −

.

Show that the Chebyshev polynomials { } { }2

1 2, , 2 1T T x x= − are orthogonal with inner product

NOTE: 1

2 22(1 ) 1x x dx x−

− = − −∫ ,

)2 2 1 2(1 ) sin ( ) 1x x dx x x x− = − − ,

2 21

3 2 2( 2) 1

(1 )3

x xx x dx

− + −− = −∫

1/20/2012

is the solution to the Least Square Minimization problem

(the same as obtained by using projections or Best

Schmidt

are orthogonal with inner product

2 2( 2) 1

3

x x+ −

Additional Exercises Handout, Math 344 Al Jimenez, Cal Poly 1/20/2012

For Section 11.2

Use Fourier

series to

calculate yp