Differential equations Questions and Answers

Solve the given initial value problem y x y 18y 81y 0 y 0 0 y 0 1 y 0 2

Solve the given initial value problem y x y 49y 0 y 0 3 y 0 6

5 15 points USING ANALYTICAL TECHNIQUES for the function given below do the following f x frac 7 x 3 2 x 2 7 x 2 16 x 4 a Find the open intervals where the function is increasing and decreasing Note If you draw a graph and find the write the intervals the answer is correct and show NO ANALYTICAL work you get 2 points b Find the slant asymptote Show all STEPS

4 The half life of extra strength Tylenol is 3 hours Suppose you take 1000 milligrams of Tylenol Extra Strength find a model which expresses the amount of Tylenol Extra Strength remaining in your bloodstream t hours since you took it

Find if f x 4x 4 16 4x 1 and 0 8

Find the critical point of the function f x y 12x 6y In x yl C Use the Second Derivative Test to determine whether it is A test fails B a local minimum C a saddle point D a local maximum Show Transcribed Text

4 Cons the system begin array i frac d x d t y x y 1 frac d y d t x y x end array a Find the and X Y nullclines of the system Sketch and label the nullclines on the graph below Be sure to indicate the direction of the solution on the nullelines b Find all of the equilibrium solutions for the system Label these on the graph below c Sketch the trajectory of a solution curve starting at x 0 1 and y 0 on the graph below 1

reflect C A R if C 1 4 A 2 3 and R 3 2 over the line y x

Simplify the trigonometric expression 1 sin u cos u cos u 1 sin u

t x o t t X t 1 4 olves the initial value problem X AX F t X t X whenever t is a fundamental matrix of the associated omogeneous system Use the above to solve the given initial value problem s F s ds X x 0 6e6t

Fran Bowen created the following budget Food 356 Clothing Transportation 379 Housing 966 Food Personal expenses and recreation She actually spent 387 for food 433 for transportation 1 074 for housing 216 for clothing and 307 for personal expenses and recreation Calculate the variance for each of these categories and indicate whether it was a deficit or surplus Input all amounts as positive values Transportation Housing Clothing Personal expenses and recreation 193 365 LA LA LA Click to select Click to select Surplus Deficit

A person starts at time t 0 to pay part of their salary continuously into an investment fund at a rate of P dollars per year The fund earns interest at a rate r per year compounded continuously The amount of money in the fund after t years M t then satisfies the differential equation dM dt P rM with M 0 0 i Solve the differential equation using the integrating factor method ii If the person can pay in 1 250 each year and the interest rate is 5 how long will it take for the amount of money in the fund to reach 10 000

5 Consider the differential equation dy dt a Sketch a slope field on the axes at right b Sketch the solutions whose initial values are y 0 4 3 4 c What do your sketches suggest is the solution whose initial value is y 0 1 Verify that this is indeed the solution to this initial value problem t y 4 3 2 1 4 3 2 1 N y 1 2 3 4 t 12 3 4

Write in factored form so the leading coefficient of the polynomial factor is positive 4a 7

me indicated function y x is a solution of the associated homogeneous equation Use the method of reduction of order to find a second solution y x of the homogeneous equation and a particular Olution y x of the given nonhomogeneous equation y y 1 2 x o x Y 1

Consider the differential equation 4y 4y y 0 ex 2 xex 2 Verify that the functions ex 2 and xex 2 form a fundamental set of solutions of the differential equation on the interval The functions satisfy the differential equation and are linearly independent since W ex 2 xex 2 Form the general solution y 0 for x 0

Caoimhe opens two bank accounts a checkings account with balance b t and a savings account with balance s t a time t measured in years since opening the accounts At the same time Caoimhe deposits 1000 dollars into the checkings account and 500 dollars into the savings account The bank is paying interest at continuous rate of 3 or the checkings account and 7 on the savings account Additionally Caoimhe continuously transfers money from the checkings account to the savings account at the rate of half the balance of the checkings account C C3 GA Set up a linear system of differential equations of the form db dt c b c 8 ds dt C3b C4S

a y y y 4 x x x 1 25 cos 2 x 11 b y 2y y 2y ex sih 2x

If you shovel 0 63 metric tonnes of dirt you shovel kilogra

Please select the answer you think is closest to being a true statement The width of a sewing needle is approximately 1 2 decimetres 5 6 metres O 1 2 millimetres 1 2 centimetres

2 2 Find a formula for the n partial sum of the series 2 3 The formula for the n partial sum s of the series is 2 and use it to find the series sum if the series convers 3 1 9 27 18 1

k 0 Set up a differential equation for the quantity Q t in milligrams of the drug in the body at time t hours Assume the are 0 3mg of drug in the bod initially Your answer will contain and r k VOG 11

f x g x 2 2x 2 2 if g 2 4 and g derivative 2 3 then find f derivative 2

1 point Use a Riemann sum to estimate the volume of the solid that lies below the surface ry and above the region R z y 0 9 0 y 8 Create the Riemann sum by subdividing 0 9 into m 3 intervals and 0 8 into n 2 intervals and take the sample point to be the upper right corner of each rectangle Volume I

Given the parabola f x x and a point 2 3 not on the parabola find a point a b on the parabola so that the tangent lines passes through both p Please show all of the steps required to find the point a b

D A 1 2 1 1 3 3 1 6 5 4 a 2 12 6 a 2 5 19 to determine the dimension and a non trivial basis of the row space rowsp A t column space colsp A and the null space N A of A

Question 5 Consider the following differential equation a Determine the equilibrium solution s of the differential equation Clearly justify your steps and show your work dy dx b Determine whether the equilibrium solution s are stable semi stable or unstable Clearly justify your steps and show your work c Identify the corresponding slope field for the differential equation above Each slope field is centered at the origin Clearly justify your selection with one or two sentence s AVAVA T THE FI 0 6y y 3 M H 14 H F 47 F 1111 77 R 11124 11

Research indicates that people underestimate their personal probability of encountering negative events Suppose you wanted to conduct a study to see how people s judgments of their likelihood to be involved in an accident relates to the number of accidents they have actually been involved in Below is the number of actual auto accidents for participants who thought they were very likely and very unlikely to be involved in an accident Number of accidents for those who believed they were extremely likely to be in an accident 2 10 10 1 9 10 3 10 3 9 5 4 2 5 9 Number of accidents for those who believed they were extremely unlikely to be in an accident 4 13 0 554 433 3 3 7 9 0 95 10 a Compute the mean number of accidents for the people in the extremely likely condition b Compute the variance and standard deviation for the extremely likely condition

A plane flying horizontally at an altitude of 1 5 miles and at a speed of 704 mi hr passes directly over a radar station Find the rate at which the distance from the plane to the station is increasing when the distance between the plane and the station is 8 8 miles If necessary round your answer to two decimal places y I Variable t x Y N dx dt dy dt dz dt Description Time Horizontal distance the plane has traveled since time t Distance between the plane and the ground at time t Distance between the plane and the radar station at time t Rate at which a changes with respect to time Rate at which y changes with respect to time Rate at which z changes with respect to time The rate at which the distance is increasing is about mph Unit hr mi mi mi mi hr mi hr mi hr

Newton s Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and the surrounding medium Suppose an object is taken from an oven at 145 C and left to cool in a room at 21 C After 0 5 hours the object s temperature is 94 C Let T t denote the object s temperature after t hours Complete the differential equation that describes this situation by entering a suitable expression involving T Answer dT dt Find the object s temperature after 2 hour s Answer T 2 C h C

Imagine dropping a steel ball bearing into a tall cylinder full of motor oil Viscous drag has a significant effect on the ball bearing s downward velocity v t One family of dynamic models for this takes the form g kv where g is the acceleration of gravity and k is a constant The size of k is related to the viscosity of the medium when k 0 equation is the familiar friction free approximation for a falling mass The exponent p is an interesting topic for dicussion for this problem we consider p 1 du dt In this problem we use SI units so t is in seconds v is in metres per second etc Use the international standard value g 9 807 and assume motor oil has viscosity parameter k 18 a Find v t Assume v 0 0 v t b Let y t denote the distance the ball bearing has dropped at time t This terminology entails y 0 0 and y t 0 for t 0 Find y t y t c Using computer assistance as appropriate find the time required for the ball bearing to drop 1 metre from its release point y t 1 when t B

Find the function y y x given dy da y cos x 4 cos x Answer y x y 0 6

x 16y a Find F to express your solution in the implicit form F x y C CER Solve the differential equation dy da Answer F x y Note Infinitely many choices for F have the required property To satisfy WeBWork enter one that also makes F kx ky k F x y for all real values k x y b Express the solution whose graph includes the point x y 4 1 as an explicit function y y x Enter the whole equation including y on the left Answer c Express the solution whose graph includes the point x y 0 4 as an explicit function y y x Enter the whole equation including y on the left

du Find the function u u t given that u 0 0 and dt 5te u

Work through the suggested steps to solve the initial value problem ODE sec z y2 cos z y 0 2 IC y 0 3 3 a Separating variables and integrating leads to 9 1 dy f 2 dz where g y and f x Note Choose functions f and g for which f 0 1 You can arrange this if necessary by multiplying both sides of whatever integral identity you derive b a well chosen constant 813 b Evaluating the integrals above gives G y F x R where R is an arbitrary constant formed by combining the constants for the indefinite integrals G y g u dy and F x f x dx c The solution to the given initial value problem is

Consider the growth of a population p t It starts out with p 0 A Suppose the growth is unchecked and hence p kp for some constant k Then p t Of course populations don t grow forever Let s say there is a stable population size Q that p t approaches as time passes Thus the speed at which the population is growing will approach zero as the population size approaches Q One way to model this is via the differential equation p kp Qp p 0 A The solution of this initial value problem is p t

The differential equation below has infinitely many solutions dy 8y da x All solutions can be expressed in the form F x y C for some function F that satisfies F 1 1 1 Find this F Answer F x y 1 x 0 Note The question as stated has infinitely many correct answers for F but WeBWork is looking for a specific one whose form is F x y r y integers a and b with b 0

legend for yourself of the terms learnec in this section of f x x 3 3 5 sin2x e 5x

The weight of a population of yeast is given by a differentiable function y where y t is measured in grams and tis measured in days The weight of the yeast population increases according to the equation ky where kis a constant At time t 0 the weight of the yeast population is 120 grams and is increasing at the rate of 24 grams per day Which of the following is an expression for y t dy A B C D 120241 120V 5 e 5 119 241 120

nd an f x and a g x such that h x fog x 1 x 1 3 h x 22 f x g x

2 Consider the continuous time system given by d x t dt y t 2 3 3 x t 2 y t a Find the general solution for the system of differential equation b Draw a phase portrait for the system

Express the following angular speed in radians per second 9 revolutions per second The angular speed is radians per second

is the region that is bounded by the spherical surface x y x 9 and the cones and for z x y 3x x y z 0 i Sketch the projections of solid G onto the three coordinate planes 6 text marks ii In spherical coordinates write a triple integral that gives the volume of solid G

Find the inverse Laplace transform L F s of the given function 8s 18s 216 F s L F s s s 36 Your answer should be a function of t

Explain whether the following statement is true or false If y t is a solution of y sin t y t y 0 whose graph in the ty plane touches and is tangent to the t axis at to y t is the constant zero 0 then

Score 4 10 4 10 answered Question 10 5 4 3 2 1 Write an equation for the polynomial grap 6 5 4 3 2 1 1 2 3 4 5 1 2 3 4

A cumulative percent of Americans in each credit score range as of April 2021 is summarized with the Ogive curve If the minimum credit for mortgage the care is 620 which minimum percentile should the potential buyers fall in Use the Ogive 20th 30th 50th or 70th use numbers only 100 0 BOOK 600 400 200 0 0 20 O Cum Rel Frequency 200 400 600 800

Show that the given differential equation is not exact and solve it by finding an appropriate integrating factor 6xy dx 4y 9x dy 0

X 2 d u d x X 4u du dr xsin log x Show Transcribed Text

Find the critical points of f x y 32 The four critical points are type your answer Enter Ordered Pairs 6y 1 0 2 0 0 type your answer type your answer