Solve y’’ – 11y’ + 24y = e^x +3x using reduction of order

A binary string is a “word” in which each “letter” can only be 0 or 1

Prove that there are 2^n different binary strings of length n.

Note:

• Your goal is to produce a properly constructed proof by induction, but this does not mean you have to follow
• Mathematical induction, step-by-step..
• Write the statement with n replaced by k
• Write the statement with n replaced by k+1.
• Identify the connection between the kth statement and the (k+1)th statement.
• Complete the induction step by assuming that the n=k version of the statement is true, and using this assumption to prove that the n=k+1 version of the statement is true.
• Complete the induction proof by proving the base case.
• point-by-point.
• ANOTHER IMPORTANT NOTE:
• Make sure you are addressing the given statement: "there are 2ⁿ different binary strings of length n” !!! So your solution should primarily be discussing binary strings. Yes, the fact that 2^k+1 = 2^k∙2 will be useful in your solution, but it should not be the sole content of your solution, as by itself that equality is a fact about exponents, not a fact about binary strings.

1. Average personal income in Hawaii increased approximately linearly from $41.7 thousand in 2010 to $50.6 thousand in 2016. Average personal income in Colorado increased approximately linearly from $39.9 thousand in 2010 to $52.1 thousand in 2016 (Source: U.S. Bureau of Economic Analysis).

   1. Let H(t) be the average personal income in Hawaii and C(t) be the average personal income in Colorado, both in thousands of dollars, in the year that is t years since 2010. Find equations of H and C.

   2. Use substitution or elimination to estimate when average personal income in Hawaii was equal to average personal income in Colorado. What is that average personal income?

   3. Use a graphing calculator table or graph to verify the result you found in part (b).

   4. Find the average of H(11) and C(11). Assuming the population of Colorado will continue to be larger than the population of Hawaii, will your result likely be an underestimate or an overestimate of the 2021 average personal income of residents of Hawaii and Colorado put together? Explain.

Show that if V is finite-dimensional and W is infinite-dimensional, then V and W are NOT isomorphic.

1. How many ways can a Snail Darter Society, who has 25 members, elect an executive committee of 2 members, an entertainment committee of 3 members, and a welcoming committee of 2 members if members can serve on at most 2 committees?

6) (8 pts, 4 pts each) State the order of each ODE, then classify each of them as
linear/nonlinear, homogeneous/inhomogeneous, and autonomous/nonautonomous.
A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 0

B) Simple RLC Circuit with a 9V Battery: Lq′′ + Rq′ +(1/c)q = 9

7) (8 pts) Find all critical points for the given DE, draw a phase line for the system,
then state the stability of each critical point.
Logistic Equation: y′ = ry(1 − y/K), where r < 0

8) (6 pts) A mass of 2 kg is attached to the end of a spring and is acted on by an
external, driving force of 8 sin(t) N. When in motion, it moves through a medium that
imparts a viscous force of 4 N when the speed of the mass is 0.1 m/s. The spring
constant is given as 3 N/m, and this mass-spring system is set into motion from its
equilibrium position with a downward initial velocity of 1 m/s. Formulate the IVP
describing the motion of the mass. DO NOT SOLVE THE IVP.

9) (8 pts, 4 pts each) Find the maximal interval of existence, I, for each IVP given.
A) (t^2 − 9) y′ − 7t^3 =√t, y(−2) = 12

B) sin(t) y′′ + ty′ − 18y = 1, y(4) = 9, y′(4) = −13


10) (30 pts, 10 pts each) Solve for the general solution to each of the DEs given. Use an
appropriate method in each case.
A) Newton’s Law of Cooling: y′ = −k(y − T)

B) (sin(y) − y sin(t)) dt + (cos(t) + t cos(y) − y) dy = 0

C) ty′ − 5y = t^6 *e^t

11) (30 pts, 10 pts each) Solve for the general solution to each of the DEs given, then
classify the stability and type of critical point that lies at the origin for each case.
A) y′′ + y′ − 132y = 0

B) y′′ + 361y = 0

C) y′′ + 6y′ + 10y = 0

12) (10 pts) Solve for the general solution to the DE given.
y′′ − 9y = −18t^2 + 6

Computing3√25 using MATLAB.

(a) Beginning with the interval [2,3], and f(x) =x^3−25, use the error bound for bisection to predict how many iterates bisection would need to get error within 10^−20.

(b) Run bisection on this problem. How many iterations did it need? For some of the iterates compute the absolute error. What is happening approximately to the number of significant digits of accuracy with each iteration?

(c) Write a program to perform Newton’s method on f(x) =x^3−25 with p0= 3 to get an approximation to 3√25 that is accurate to within 10^−20. Calculate the derivative by hand and hard code it in as a new function. How many iterates did it need? Save all of the iterates.

(d) Compute the absolute error and the number of significant digits of accuracy for each iterate. What is happening to the number of significant digits with each iteration? Compare to bisection

Prove or disprove: If a, b, c are any three distinct positive integers such that 1/a + 1/b + 1/c = 1, then a + b + c is a prime

Suppose that a set G has a binary operation on it that has the following properties:

1. The operation ◦ is associative, that is:

for all a,b,c ∈ G, a◦(b◦c)=(a◦b)◦c

2. There is a right identity, e:

For all a∈G a◦e=a

3. Every element has a right inverse:

For all a∈G there is a^-1 such that a◦a^-1=e

Prove that this operation makes G a group. You must show that the right inverse of each element is a left inverse and that the right identity is also a left identity. See how many proofs you can give.

If G is a group, show that the set of automorphisms of G and the set of inner automorphisms of G are both groups.

Find the first five nonzero terms in the solution of the given initial value problem.

y′′−xy′−y=0, y(0)=3, y′(0)=4

Enter an exact answer.

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x, y) = 5(x2 + y2)ey2 − x2 local maximum value(s) Correct: Your answer is correct. local minimum value(s) Incorrect: Your answer is incorrect. saddle point(s) (x, y, f) =

Use the following Venn diagram tools to construct diagrams that represent the given syllogistic forms from the Boolean standpoint. Once you have constructed your diagrams, interpret them to determine whether each syllogistic form is valid or invalid from the Boolean standpoint. Indicate your answers using the dropdown menus below each diagram.

Argument 1

This syllogistic form is ________ from the Boolean standpoint.

Argument 2

This syllogistic form is ________from the Boolean standpoint.

Argument 3

This syllogistic form is ________from the Boolean standpoint.

Argument 4

UIC

This syllogistic form is ________ from the Boolean standpoint.

Students in a gym class have a choice of swimming or playing basketball each day. Thirty percent of the students who swim one day will swim the next day. Sixty percent of the students who play basketball one day will play basketball the next day. Today, 100 students swam and 200 students played basketball. How many students will swim tomorrow, in two days, and in four days? (Round your answers to the nearest whole number.)

(a) tomorrow: _____students

(b) two days: _____students

(c) four days: _____students

a) ? 2 − ? = 49 b) ? 2 + ? − ? 2 = 49 c) ? 2 + ? 2 − ? 2 = 49 d) ? 2 − ? 2 − ? 2 = 49 e) ? 2 + 2? 2 + ? 2 = 49 f) ? 2 + ? 2 − ? = 49 7. Which of the given equations represent a cylinder surface? What kind of cylinder is it, and what axis is it parallel to? 8. What are the traces of the graph of equation (d) in the planes ? = 6, ? = 7 and ? = 8? (Give equations in ? and ?, and name the 2D shapes.) What kind of 3D surface is the graph of equation (d)