Differential equations Questions and Answers

es the graph represent a function that has an verse function 10 10 10 10 Q Q

1 Consider the family of curves described by x xy ay b Remember to justify your answers A Let a 1 and b 7 Find the two x intercepts B Let a 1 and b 7 Using the two points you found in part A show that the tangents to the curve at these points are parallel and find the common slope D

Find a unit vector u in the direction opposite of 3 9 10 u

Distance and Dot Products Consider the vectors Compute u Compute v Compute u v u 4 0 10 and v 2 1 9

Consider a triangle ABC like the one below Suppose that A 25 C 117 and a 39 The figure is not drawn to scale Solve the triangle Round your answers to the nearest tenth If there is more than one solution use the button labeled or B 0 b 0 or X No solution S

Use the information given about the angle 0 0 0 2x to find the exact value of cos 20 3 cos8 7 csc8 0 OA OB O c O D 12 10 49 31 49 12 10 49 Question 8 of 12 31 49

A Suppose the height of an object at any time 20 is given by the function h t where his feet and is in seconds The first derivative h t represents the rate of change in height with respect to time which is actually the velocity of the object and the units are feet per second What does the second derivative h t represent and what are the units B Suppose the population of a bacteria culture is given by the function p t where is population in thousands and represents time in hours Then p t represents the rate of change in the population of the bacteria with respect to time and its units are in thousands of bacteria per hour What does p t represent and what are its units C Suppose that 7 t is a twice differentiable function of 1 where T is in C and is in minutes What are the units of T t and T t

4 Consider the function x x x 3 x2 As in the last problem find f x showing all work and use the graph of this function to estimate where fx is concave up and where fx is concave down You don t need to include sketch of the graph of f x in your answer

Find the recurrence relation for the coefficients of the power series solution of the differential equation y y 2y 0 Assume the form y x 0 Cnx in y x n 1 Then compute First compute Cnx 1 0 y x 1 Then y y 2y Cn 2 Cn 1 Requiring that the terms of this series for y y 2y vanish gives the recurrence relation Cn 2 Cn for n 0 1 2 Cn 1 Cn 1X Cn 1X 1 0 Cn 2X Cn x

1 point For the system of differential equations x t x y 2xy 20 y t 8x 2y xy the critical point xo yo with xo 0 yo 0 is xo Change variables in the system by letting x t xo u t y t yo v t The system for u u is u v Use u and u for the two functions rather than u t and u t For the u v system the Jacobian matrix at the origin is A You should note that this matrix in 1 Yo

1 point Find the eigenvalues and corresponding eigenvectors for the matrix P A V saddle center point ellipses spiral source spiral sink none of these 28 12 6 12 and For the linear system of differential equations Px the critical point is best described as source unstable node sink stable node U

a Find the eigenvalues and corresponding eigenvectors for the matrix P A x t x 1 V 17 20 15 18 and A 11 E b Find the most general solution to the linear system of differential equations Px Use c and c to denote arbitrary constants and enter t as c1 and c2

7 Use a half angle formula to find the exact value of tan 8 tan 7z 8 2 16

Solve the following boundary value problem Ytt 25yxx 0 x 4 t 0 y 0 t y 4 t 0 y x 0 0 y x 0 x 4 x y x t Type a series using n as the index variable and 1 as the starting index

Find the derivative of the function Do not find the product before finding the derivative y 4x 3x5 5x Choose the correct answer below OA y 4x 15x4 10x 3x5 5x 8x OB y 5x 4x 3x5 5x 15x4 10x OC y 3x5 5x 15x 10x 8x 4x OD y 4x 3x5 5x 15x4 10x 8x

Find all the higher derivatives of the following function y 6x 4x dy dx d y 2 dx 11 18x 8x

Differentiate the function f x 2x 3 with respect to x

Example 6 If z f x y has continuous second order partial derivatives and x r cos a y r sin 0 show that 8 x 8 8 dy r 10 z r 20 1 z r r

Find a particular solution to the following differential equation Do NOT solve the coefficients y 4 4y 3 7y 6y 2y et 4t sint

Use Taylor series to evaluate the lim 1 In 1 8x cos 8x x lim X 0 O A 32 O B 68 O c 4 O D 40

Find the mass and center of mass of the solid E with the given density function p E is bounded by the parabolic cylinder z 1 y and the planes x 4z 4 x 0 and z 0 p x y z m x y z X

True or false y x cosa is a solution of the differential equation y y 2 sin True False

Solve the differential equation 2yy xe Please show your work to receive full credit

A model rocket is launched directly upward at a speed of 20 meters per second from the ground The function f t 4 91 20t models the relationship between the height of the rocket and the time after launch t in seconds When in seconds after launch will the rocket reach its highest point Round to two decimal places Provide your answer below seconds

56 Points DETAILS SMITHNM13 13 3 025 ppose that the odds in favor are 16 to 1 that a man will be bald by the time he is 60 State this as a probability Enter the probability as a ction man is bald by 60

1 point Convert the nonhomogeneous differential equation y 3y 7y 1 to a first order system Let x y x y Enter x and x in your answers as x1 and x2 x t x t help formulas help formulas

1 point Convert the system of second order differential equations to a first order system Let x x x x x3 y x4 y x5 Z x6 Z Enter x x2 X6 in your answers as x1 x2 x6 x t x t x t x4 1 x s t x t x 3x y 2z y x y 4z z 5x y z help formulas help formulas help formulas help formulas help formulas help formulas

1 point Find the solution to if x 0 3 and y 0 4 x t y t x y x t y y help formulas help formulas

1 point Consider the system of differential equations Verify that 3 0 X 1 e 1 is a solution to the system of differential equations for any choice of the constants c and c2 Find values of c and c that solve the given initial valu problem According to the uniqueness theorem you have found the unique solution of x Px x 0 o t DH e4r

point This is the first part of a two part problem ewrite the given system of linear homogeneous differential equations as a homogeneous linear system of the form y Py y 2y1 y2 Y3 y y2 2y3 1 y 2y2 y3 Y Y1 3 2 Va

Test whether the Gauss Seidel iteration converges for the system 0237 the 2x x x3 4 X 2x x3 4 x x 2x3 4 Use a suitable norm in your computations and justify the choice

If labor x costs 260 per unit materials y cost 123 per unit and capital z costs 82 per unit write a function for total cost G XYZ www

Given that graph of f passes through the Point 4 69 that the slope of HS fargent line at x f x is 10x 4 find f 1 8 b

Problem 88 Revenue and profit functions A company manufactures 10 and 3 speed bicycles The weekly demand and cost functions are p 230 9x y q 130 x 4y C x y 200 80x 30y where p is the price of a 10 speed bicycle q is the price of a 3 speed bicycle x is the weekly demand for 10 speed bicycles y is the weekly demand for 3 speed bicycles and C x y is the cost function Find R 10 5 and Pr 10 5 and interpret the results in words

QUESTION 15 7 points Find the integrals Must show all work including all steps for and Sub a 2r 3 dr b Szln 2 c r dr 1 r dr 4

Use the to Give Homogeneous Substitution method solve the following differential Equation a general xy y x F box solution

5 Use the Method of Coefficient to the Undetermined general to the following differential equation Give the comple following non homogeneous genenal y y 6 cos x Solution Solution

Of differential equations following homogeneous Using the characterist equation Give the general Show of your Wonk all y qy 0 3y 10y 0 44 54 the solution

3 Use the integration factor Method to fir the general solution of y xy 4x

2 Use the Following Give the YE separation Separable Particular x ex technique to solve th equation differential solution y 0 3

fine Set up and solve a Definite integral to the particular solution of the following equation must Use Propert differential integral with all the a Definite notation AISO Bounded integral 0 6 must use dy 4x 6x ex dx

Given matrices A and B shown below find 6 A 9 B B 2 5 1 0 3 3 3 2 4 0 1 1 0 Give your answer as a matrix using the matrix tool A 3 b n b 3 L 5 T

Given matrix A shown below find 1A Give your answer as a matrix using the matrix tool 2 3

ADVANCED ANALYSIS Assume that the consumption schedule for a private open economy is such that consumption C 50 0 75Y Assume further that planned investment g government spending G and net exports Xn are independent of the level of real GDP and constant at g 30 G 0 and X 10 Recall also that in equilibrium the real output produced Y is equal to aggregate expenditures Y C Ig G Xn Instructions Round your answers to the nearest whole number a Calculate the equilibrium level of income or real GDP for this economy b What happens to equilibrium Y if lg changes to 10 What does this outcome reveal about the size of the multiplier Multiplier

x 8 sin 4x cos 4x c1 cos 4x c2 sin 4x c1 cos 4x c2 sin 4x x sin 4x 8 x cos 4x 8 t least one of the answers above is NOT correct c cos 4x c sin 4x c cos 4x c sin 4x sin 4x cos 4x 1 point a Find a particular solution to the nonhomogeneous differential equation y 16y cos 4x sin 4x help formulas help formulas help formulas x sin 4x 8 x cos 4x 8 correct yp x 8 sin 4x cos 4x b Find the most general solution to the associated homogeneous differential equation Use c and c in your answer to denote arbitrary constants Enter c as c1 and c as c2 Yh c1cos 4x c2sin 4x c Find the solution to the original nonhomogeneous differential equation satisfying the initial conditions y 0 5 and y 0 2 y c1cos 4x c2sin 4x xsin 4x 8 xcos 4x 8 correct incorrect

Problem 3 6 points Compute the rate of change of the function f x y ln xy in the direction of k 1 at the point 1 1

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions y 7y 6y et 8 t 3 8 t 7 y 0 0 y 0 0 1 u t 3 y t X X X u t 7

ind all solutions to the differential equation xy y x y 0 x 0

ompute the following limits using I H opital s rule if appropriate Use INF to denote and MINF to denote x mx 0 X 3 512 2 3 dt

complete a Sketch at least 2 cycles of the graph showing the height of a rider above the ground as a function of time if the rider gets on at a height of 0 5 metres at t 0 minutes 4 marks b Determine the equation which expresses the height of the rider with respect to time Show all calculations Use radian measure 5 marks