Calculus HW

mirananwar

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Miran Anwar, mth 235 ss21 1: Hw11-3.3-LT-DS. Due: 04/26/2021 at 11:00pm EDT.

See in LN, § 3.3, See Example 3.3.11. 1. (10 points)

(a) Find the function f such that L [

f (t) ] =

5 s e−2s

3 s2 + 12 .

f (t) =

(b) Find the function g such that L [ g(t)

] =

4 e−s

5 s2 + 125 .

g(t) =

Note: If you need the step function at t = c, it should be entered as u(t −c).

See in LN, § 3.3, See Example 3.3.12. 2. (10 points)

(a) Find the function f such that L [

f (t) ] =

4 3(s + 1)2 + 48

f (t) =

(b) Find the function g such that L [ g(t)

] =

4(s + 1) 2(s + 3)2 + 18

g(t) =

Note: If you need the step function at t = c, it should be entered as u(t −c).

See in LN, § 3.3, See Example 3.3.12. 3. (10 points)

(a) Find the function f such that L [

f (t) ] =

4 2(s + 2)2 −32

f (t) =

(b) Find the function g such that L [ g(t)

] =

4(s−2) 3(s−1)2 −27

g(t) =

Note: If you need the step function at t = c, it should be entered as u(t −c).

See in LN, § 3.3, See Example 3.3.10. 4. (10 points) Consider the function

f (t) =

{ 0 for t < 3,

3t 2 + 2t −2 for t > 3.

(a) Use the step function at t0, denoted as u(t −t0), to rewrite the function f as

f (t) = u(t −t0) ( A(t −t0)2 + B(t −t0)+C

) ,

Find the constants t0, A, B, and C.

t0 = A =

B = C =

(b) Find F = L[ f ], the Laplace transform of f .

F(s) =

See in LN, § 3.3, See Example 3.3.10. 5. (10 points) Consider the function

f (t) =

 

0 for t < 5π 6 ,

3 sin(6t) for t > 5π 6 .

(a) Rewrite the function f as follows:

f (t) = A u(t −t0)sin(6(t −t0)).

Find the constants t0 and A.

t0 = A =

(b) Find F = L[ f ], the Laplace transform of f .

F(s) =

See in LN, § 3.3, See Example 3.3.15. 6. (10 points)

Consider the initial value problem for function y given by,

y′′−5 y′+ 6 y =−4 u(t −3), y(0) = 0, y′(0) = 0.

where u(t −c) denotes the step function with step at t = c. 2

Part 1: Finding F(s) (a) Find the Laplace Transform of the source function, F(s) = L

[ −4 u(t −3)

] .

F(s) =

Note: We are asking only for the Laplace Transform of the right side of the differential equation; not for the Laplace Transform of the solution.

Part 2: Finding Y (s) Part 3: Rewriting Y (s) Part 4: Finding y(t)

See in LN, § 3.3, See Examples 3.3.15, 3.3.16. 7. (10 points)

Consider the initial value problem for function y given by,

y′′−2 y′−8 y = 5 u(t −4)e3t, y(0) = 0, y′(0) = 0.

where u(t −c) denotes the step function with step at t = c.

Part 1: Finding F(s) (a) Find the Laplace Transform of the source function, F(s) = L

[ 5 u(t −4)e3t

] .

F(s) =

Note: We are asking only for the Laplace Transform of the right side of the differential equation; not for the Laplace Transform of the solution.

Part 2: Finding Y (s) Part 3: Rewriting Y (s) Part 4: Finding y(t)

See in LN, § 3.3, See Example 3.3.17. 8. (10 points)

Consider the initial value problem for function y given by,

y′′−4 y′−5 y = −u ( t − π6

) sin(6t), y(0) = 0, y′(0) = 0.

where u(t −c) denotes the step function with step at t = c.

Part 1: Finding F(s) (a) Find the Laplace Transform of the source function, F(s) = L

[ −u ( t − π6

) sin(6t)

] .

F(s) =

Note: We are asking only for the Laplace Transform of the right side of the differential equation; not for the Laplace Transform of the solution.

Part 2: Finding Y (s) Part 3: Rewriting Y (s) Part 4: Finding y(t)

See in LN, § 3.3, See Example 3.3.10. 9. (0 points) Consider the function

f (t) =

 

0 for t < 5π 4 ,

5 cos(4t) for t > 5π 4 .

(a) Rewrite the function f as follows:

f (t) = A u(t −t0)cos(4(t −t0)).

Find the constants t0 and A.

t0 = A =

(b) Find F = L[ f ], the Laplace transform of f .

F(s) =

See in LN, § 3.3, See Example 3.3.17. 10. (0 points)

Consider the initial value problem for function y given by,

y′′−3 y′−4 y = 3 u ( t − π6

) cos(6t), y(0) = 0, y′(0) = 0.

where u(t −c) denotes the step function with step at t = c.

Part 1: Finding F(s) (a) Find the Laplace Transform of the source function, F(s) = L

[ 3 u ( t − π6

) cos(6t)

] .

F(s) = 4

Note: We are asking only for the Laplace Transform of the right side of the differential equation; not for the Laplace Transform of the solution.

Part 2: Finding Y (s) Part 3: Rewriting Y (s) Part 4: Finding y(t)