Differential equations Questions and Answers

Homework 2 1 Elementary De core 0 16 0 15 answered Question 12 10 4 5 3 2 5 4 3 2 1 1

If the x axis is translated 10 places up and the y axis 4 places to the right the equation x 4 10 1 changes to which of the following 25 9 A x 4 3 10 S D 7 0 3 B x 4 x 10 3 E x 1 3 5 C 21 x y 25 9

Y 1 9972x 1 365x 2 3 2702x 1 365 0 0132x

3 Use Math Induction to prove the following true for all positive integers 3 3 3 3 1 2 3 n n n 1

Consider the function graphed to the right The function is increasing on the interval s The function is decreasing on the interval s The function is constant on the interval s The domain of the function is The range of the function is 6 5 4 3 2 6 5 m 2 25 22 3 4 5 6 6 1 2 3 4 5 6 O

20 1emerlod sin x csc X CSC tadi obulong csc x sin x

7 Observe that r t a cos t b sin t defines an ellipse a Find the arclength of r t b Find the curveature of r t

Find the vertical horizontal hole and or oblique linear or parabolic asymptote for each function Do not graph 1 f x 3x 3x 18 x x 6x 3x 3x 18 3 f x 2 f x 2 x 4 4 f x x x 6 4x 16x 12

A Bernoulli differential equation is one of the form frac d y d x P x y Q x y n Observe that if n 0 or 1 the Bernoulli equation is linear For other values of 9 n the substitution u y n transforms the Bernoulli equation into the linear equation frac d u d x 1 n P x u 1 n Q x Use an appropriate substitution to solve the equation y prime frac 9 x y frac y 5 x 5 and find the solution that satisfies y 1 1 y x

Find sin a and cos tan a and cot and sec a and csc a a sin a and cos p b tan a and cot c sec a and csc 20 B 12

Determine whether the integral is divergent or convergent If it is convergent evaluate it If it diverges to infinity state your answer as INF If it diverges to negative infinity state your answer as MINF If it diverges without being infinity or negative infinity state your answer as DIV int 2 10 frac 2 x 6 3 d x

Given the Exponential Function determine the Initial Value and Rate of Change as a Percent for each of the following Initial Value 1850 3746 4060 3421 4375 645 Growth or Decay Rate as a 400 Growth 12 Growth 6 Decay 14 Decay 25 x Growth 9 Growth VV Exponential Function p t 1850 5 f x 3746 1 12 d t 4060 0 94 f x 3421 0 86 h x 4375 1 055 g x 645 1 09

Determine an equation for the graph of a polynomial Write your answer in factored form and assume the leading coefficient is 1 y 7 6 5 3 2 1 3 4 5 6 7 a

Find the central angle 0 0 rad 0 3 7

solve the followin equation using the method of undetermined coefficients Question 2 6 6 8 8 28 Solve the following equations using the method of Undetermined Coefficients for 1 2 3 and the Variation of Parameters for 4 1 y 6y 13y 13x x

A bacteria culture initially contains 2000 bacteria and doubles every half hour Round your answers to 1 decimal place Find the size of the bacterial population after 40 minutes 5039 68 Find the size of the bacterial population after 7 hours 5040 X

2 For the polynomial function f x 7 x 4 x 4 answer the following questions a List each real zero and its multiplicity b Determine whether the graph crosses or touches the x axis at each x intercept c Determine the maximum number of turning points on the graph d Determine the end behavior that is find the power function that the graph of f resembles for large values of x

The polynomial f x 3x4 2x 4x wx 9 is divided by x 2 For what va will the remainder be 1 W

1 For each of the following parametrized family of differential equations plot phase lines for X 2 1 0 1 2 find any bifurcation values and sketch the bifurcation diagram a b for c d e y y y Ay 1 XER y x 2y y y x 4y y y x y y

1 QuchuIOITO VI m 2 The slope of a line is given 5 11 To pomy QuesuUIT ALE Part 1 of 2 a Determine the slope of a line parallel to the b Determine the slope of a line perpendicular 4 a The slope of the parallel line is

The population of bacteria in millions in a certain culture x hours after an experimental 15x nutrient is introduced into the culture is P x for the following changes in x a 2 to 2 5 2 7 x b 6 to 6 25 Use the differential to approximate the changes in populati ACCEN a Use the differential to approximate the change in population for x 2 to 2 5 Between 2 and 2 5 hours the population of bacteria changes by 0 186 million Round to three decimal places as needed b Use the differential to approximate the change in population for x 6 to 6 25 Between 6 and 6 25 hours the population of bacteria changes by million Round to three decimal places as needed

3 a 10 points Assume the characteristic polynomial of the differen qx 1 0 can be factorized to Ac 2 Justify that ect and tect form a solution set using their Wronskian b 15 points Solve the initial value problem 3 x 6x 9x 0 x 2 e x 2 5e6 7 6x 9x e

The lowest point of y is at 0 0 T Where is the lowest point of y x 1 3 O 1 3 1 3 1 3

1 12 pts Each function below has domain 0 0 and passes through 0 2 and 2 5 For each give a rough sketch of the function and then find an exinct formula a A line b An exponential function c A function of the form y D Acos Bx with minimum value at 0 2 and maximum value at 4 8

A tank is initially fill with 100 gallons of pure water Brine with a salt concentration of 10 oz gal flows into the tank at a rate of 3 gal minute The well mixed contents of the tank leaves through a valve a the same rate 1 Set up and solve the initial value problem for the amount of salt in the tank 2 What is the amount of salt in the tank as t

Problem 1 Solve the following initial value problem 16y 0 y 0 2 0 4

The radioactive substance uranlum 240 has a half life of 14 hours The amo the following exponential function A t 5600 1 2 t 14 Find the amount of the sample remaining after 11 hours and after 50 hours Round your answers to the nearest gram as necessary

1 34 V 5 4 3 2 1 1 2 3 4 5 3 The graph above shows the base of an object Compute the exact value of the volume of the object that cross sections perpendicular to the base and parallel to the y axis are isosceles right triangles their hypotenuse in the base

The state lottery board is examining the machine that randomly picks the lottery numbers On each trial the machine outputs a ball with one through 9 on it The ball is then replaced in the machine The lottery board tested the machine for 200 trials and got the following results Outcome 0 1 2 3 16 18 4 201 5 20 6 Number of Trials 28 Answer the following Round your answers to the nearest thousandths a From these results compute the experimental probability of getting an even number 0 b Assuming that the machine is fair compute the theoretical probability of getting an even number 7 24 15 17 c Assuming that the machine is fair choose the statement below that is true 8 9 18 24 X

1 Name two pairs of congruent angles in the figure

y x dx xydy 0 2 Final Answer 1 In 1 4 In x Co

Solve the differential equation x 3 y 5y The solution is y Type an expression 2sinx X 3 4 x 3

g x 0 18 a 1 10 2 6 Find f f x dx given k 16 2 5 2 1 Let f be the function given by f x 2 Let g be a continuous decreasing function with selected values shown in the table above x k 4 1

Given Problem LINEAR DE OF ORDER ONE 3xy 3y 4 dx x 1 dy Final Answer y 2 x 2x 3 2 x 3x 3x 1 0 C1 x 3x 3x 1

6x y dx y 2x 3 dy 0 y dx Final Answer f x y 2y x 3y 6x 2 C Pls show neat and whole solution also involves the exactness like how did it become exact show the value of Mdy and

cos xcos y cot x dx sin xsin ydy 0 Final Answer Final Answer

cos 2y 3x y dx dx cos 2y 2xsin 2y 2x y 2x y dy 0 3 Final Answer f x y x cos 2y y x C sin 2y 2

y 2xy 6x dx x 2xy 2 dy 0 Final answer x y x y xy 2y 3x C

Given Problem EXACT DE 2x 3x y ye x dx x 3y e x dy Final answer 0 x y x y y ye x 2x C Pls show neat and whole solution

2x ycos xy dx xcos xy dy 0 Final Answer f x y sin xy x C

x 3xx 4y 6 dx x 6x y 1 Final Answer 32 x y yx 3x y C

x y dx 3xy dy 0 Final answer 1 1 2 In 1 4 2 12 In 2 Co 4y 7 3 3

28 cos 314 40 O A 7030 OB 7112 O C 7030 OD 7112

y x y2 2 dx x dy Final answer any of this two y In Dla 2 X 2ec Y 1 X e 1 2 0 In x Co

Consider the following parametric equations a Eliminate the parameter to obtain an equation in x and y b Describe the curve and indicate the positive orientation x r 5 y r 9 4srs4 a Eliminate the parameter to obtain an equation in x and y y x 15x 75x 116 Type an equation b Describe the curve and indicate the positive orientation Type ordered pairs Simplify your answers and ending at 4 and ending at 4 OA A cubic curve falls to the left as r increases starting at B A cubic curve rises to the right as r increases starting at OC A quadratic curve falls and then rises to the left as r increases starting at D A quadratic curve falls and then rises to the right as r increases starting at THE 4 and ending at and ending at

the interest rate for a 9500 deposit accumulating to 11 998 compounded annual nterest rate is hot round until the final answer Then round to two decimal places as needed

10 Find a general solution to the following difference equation Yn 2 6yn 1 11yn 0 Answer y 11 C cos n C sin n where 0 tan 1 2 3

Please find a polynomial function You must have at least three terms and your highest term must be at least Find all information needed to graph BUT DO NOT GRAPH YOUR OWN POLYNOMIAL Your work should be detailed and easy to read I expect to see the factored form of the polynomial the zeros and multiplicity chart including touch cross the y intercept and a discussion on the end behavior

answer 6 1 1 5 1 x txu t 1 dt The blue line represents 1xt The green line represents u t 1 The yellow line represents 1x txu t 1 The red dot is the lower limit of integration The yellow dot is the upper limit of integration

y 0 02y 6 y 0 40 has solution y t