Differential equations Questions and Answers

1 point Consider the initial value problem y y 31 y 0 2 y 0 2 Suppose we know that a complementary solution is y t c cos t sin t and that a particular solution is y 3t Find a solution satisfying the given initial conditions y t help formulas

1 point Solve the initial value problem y 9y 20y 0 y t y 0 1 y 0 0

1 point Find the two values of r for which is a solution of the differential equation smaller value larger val y x ex y 18y 77y 0

Graph the function Then identify the domain range and y intercept and state whether the function is increasing or decreasing f x e 2x Graph the function Choose the correct graph below OA 12 06 2 O C 6 A 10 12 14 16 Ay 16 14 12 10 8 6 06 12 12 06 2 06 12 Q The function is 5 Q 5 The function s domain is Type your answer in interval notation Use integers or fractions for any numbers in the expression The function s range is Type your answer in interval notation Use integers or fractions for any numbers in the expression Identify the function s y intercept Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function s y intercept is Type an ordered pair OB The function has no y intercept CIES OB Ay 1 2 0 6 2 4 D 6 84 10 12 14 16 12 10 8 6 2 12 06 2 06 12 06 12 Q

Solve the given symbolic initial value problem y 6y 8y e s t 3 y 0 2 y 0 6 y t

Solve the second order equation xy 2y 8x by making the substitution u y 1 2 32 Cx 5x

Find a solution to y ke x 2 2 5x Hint dy xy 9x 5y 45 If necessary use k to denote an arbitrary constant dx help equations

1 point 1 t 2 16 dt At least one of the answers above is NOT correct 1 8 In x 4 x 4 y x 0 a Set up an integral for solving dy dx 1 8 plx 1 x 4 X t 0 1 x 16 when y 0 0 1 t 2 16 dt b Evaluate your answer to the previous part to find y x In 0 1 12 16 dt 4 x 4 CC C in

a Set up an integral for solving y x y x dy x x when y 4 10 dx t b Evaluate your answer to the previous part to find y x help formulas

dy 1 point Find a solution to dx xy 9x 5y 45 If necessary use k to denote an arbitrary constant help equations

x 3 y 0 east one of the answers above is NOT correct the questions remains unanswered point Solve the differential equation dy X dx 9y a Find an implicit solution and put your answer in the following form x 2 2 9y 2 2 constant help formulas b Find the equation of the solution through the point x y 3 1 x 3y help equations N x 3y 0 of the colution through the point x y 0 7 Your answer should be of the form y f x correc incorre

A population is modeled by the differential equation dP P 1 5P 1 4500 dt a For what values of P is the population increasing Enter your answer in interval notation PE b For what values of P is the population decreasing Enter your answer in interval notation PE c What are the equilibrium solutions Enter your answers as a comma separated list P

a For what values of r does the function y e satisfy the differential equation 5y 34y 7y 0 Enter your answers as a comma separated list r b If r and r are the values of r that you found in part a show that every member of the family of functions y ae 1 be 2 is also a solution Let 1 be the larger value and r be the smaller value We need to show that every member of the family of functions y ae 1 be 2 is a solution of the differential equation 5y 34y 7y 0 We must decide whether 5f x 34f x 7f x 0 for f x ae 1X be 2x Substitute your values for r and r into f x then find f x and f x We have f x and f x Substituting and combining like terms we get the following C De x Therefore f x ae1 be 2 is a solution to the differential equation 5y 34y 7y 0 5f x 34f x 7f x

13 Write out the Taylor series that represents the function f x x at the point a 1

Solve 8 sin t 2 cos t 5 0 for all solutions 0 t 2

Another model for a growth function for a limited population is given by the Gompertz function which is a solution of the differential equation below where c is a constant and K is the carrying capacity K dP dt cln a Solve this differential equation P 0 Po P t c 0 P b Compute t lim P t lim P t If there is no solution enter NONE in the answer box

Evaluate Sc xy y 2 ds along the curve r t 2ti tj 3 2t k 0 t 1 Sc xy y 2 ds Simplify your answer

y 1 t 2 4 t 2 4 2 C The answer above is NOT correct 1 point Find the general solution to Enter your answer as y Use C to denote the arbitrary constant in your answer y 1 t 2 4 t 2 4 2 C y help equations 1 22 4 4 y 2ty f 4 4 C

a For what values of r does the function y ex satisfy the differential equation 7y 34y 5y 0 Enter your answers as a comma separated list r b If r and r are the values of r that you found in part a show that every member of the family of functions y ae 1 be 2 is also a solution Let r be the larger value and r be the smaller value We need to show that every member of the family of functions y ae 1 be 2 is a solution of the differential equation 7y 34y 5y 0 We must decide whether 7f x 34f x 5f x 0 for f x ae 1 be 2 Substitute your values for r and r into f x then find f x and f x We have f x Substituting and combining like terms we get the following 7f x 34f x 5f x and f x Need Help Therefore f x ae 1 be 2 is a solution to the differential equation 7y 34y 5y 0 Read 11 Watch It Master It

x y 2 ds where C is parmeterized by Find the value of integral t 8 sin t 2 7 cos t 6 cos t for 0 t

Consider the following graph 0 The function in the graph is a solution of one of the following differential equations A y 1 xy B y 2xy C y 1 2xy Decide which is the correct equation and justify your answer For points in the first quadrant the differential equation y 1 xy Select graphed Select in the first quadrant and For points in the first quadrant the differential equation y 2xy Select graphed Select in the first quadrant and For points in the first quadrant the differential equation y 1 2xy Select graphed Select in the first quadrant and To summarize the graph is the solution to which of the three differential equations O A y 1 xy OB y 2xy OC y 1 2xy Select Select Select MY NOTES v Since the For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation Since the fur in the second quadrant this differential equation Select be the correct equation Since the For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation For points in the second quadrant this differential equation Select ASK

A tank contains 2 000 L of brine with 16 kg of dissolved salt Pure water enters the tank at a rate of 20 L min The solution is kept thoroughly mixed and drains from the tank at the same rate a How much salt is in the tank after t minutes kg yo b How much salt is in the tarik after 30 minutes Round your answer to one decimal place y kg

Solve the differential equation de t sec 0 dt Beta

A plane flying horizontally at an altitude of 2 mi and a speed of 600 mi h passes directly over a radar station Find the rate at which the distance from the plane to the station is increasing when it is 3 mi away from the station

Integrate f x y z The integral of f is x y z 2 2 x y z over the path r t ti tj 2t k 0 ast b

c1 e 1 16 c2 e 4 t 3 c1 e 1 12 c2 e 4 t 4 c1 e 1 c2 e 4 t 2 5 3 e 1 7 e 4 t 3c1e 1 12c2e 4t b Find x 1 if x 0 4 and x 0 2 x t 2 5 3e t 7e 4t c e 16c e4 3c e 12c e4 help formulas 4 c e c e 4 c1e t c2e 4t 7e4 correct least one of the answers above is NOT correct point Suppose x c e c e t a Verify that x c e c e is a solution to x 3x 4x 0 by substituting it into the differential equation Enter the terms in the orde given Enter c as c1 and c as c2 c1e t 16c2e 4t 0 help formulas correct correct incorrect

1 point Suppose x c e a Verify that x cje c e4 is a solution to x 3x 4x 0 by substituting it into the differential equation Enter the terms in the order given Enter c as c1 and c as c2 0 help formulas b Find x 1 if x 0 4 and x 0 2 help formulas

1 point Suppose y c cos kx c sin kx where k 0 is a constant and c and c are arbitrary constants Find the following Enter c as c1 and C as c2 a b dy dx d y dx c Rewrite your answer to the previous part in terms of y d y dx d Find the general solution to help formulas d y dx help formulas help formulas 9y Enter your answer as an equation y help equations

4 U U 3 2 5 2 1 TI 2TT 3TT 4TV 5T 1 100 n 2 Note Enter your answer in cosecant form Provide your answer below k 2TT TT 0

Find a basis of solutions by the Frobenius method y x 1 y 0

Given the side length a 10 and the angle A 38 on the triangle below find the lengths of b and c and the measure of angle B Do not round during your calculations but round your final answers to one decimal place Provide your answer below B 10 C b 38 A

Given the triangle below find the angle 0 in degrees Round to one decimal place 8 0 7

olve the following initial value problem y 4y 5y t 2 8 t y 0 1 y 0 2

Find each equilibrium solution x t x of the given second order differential equation x f x x 0 Use a computer or graphing calculator to construct a phase portrait and direction field for the equivalent first order system x f x y Thereby ascertain whether the critical point 0 looks like a center a saddle point or a spiral point of t system x x x 0 Comp The equilibrium solution s is are x t Type an integer or a fraction Use a comme to separate answers as needed

Find the area between the curves y e 0 3 and y 2 2x 1 from x 0 to x 1 Area

By about how much will g x y z x x cos z y sinz y change if the point P x y z moves from Po 1 1 0 a distance of ds 0 1 unit toward the point P 1 1 2 dg Type an integer or decimal rounded to four decimal places as needed

Show your work as you determine whether the sequence or series converges or diverges 8 1 n 1 O Converges O Diverges 2n 3 3n 4

Determine if convergent or divergent If convergent state whether conditionally convergent or absolutely convergent 1 n 1 O Divergent 1 n Conditionally Convergent Absolutely Convergent

Sketch several solution curves in the phase plane of the system of differential equations dx dt Ax using the given eigenvalues and eigenvectors of A A 2 A 4 V H V2 A

Given that cot 0 and 0 is in Quadrant II find sec 0 and csc 0 12 Provide your answer below

Find a z score such that 63 of the area under the standard normal curve is below that score Click the icon to view the area under the standard normal curve table

A university is trying to determine what price to charge for tickets to football games At a price of 24 per ticket attendance averages 40 000 people per game Every decrease of 52 adds 10 000 people to the average number Every person at the game spends an average of 5 00 on concessions What price per ticket should be charged in order to maximize revenue How many people will attend at that price What is the price per ticket What is the average attendance people

For the given differential equation y 0 0 5 initial condition and y 0 4 value is desired to be calculated by Heun s method Which of the following is correct for the calculation step dy dx y x 1 k2 1 1 dir Two iterations are required for the solution y 0 4 1 188 dir y 0 4 1 36 d r k1 1 25 dir k2 1 94 d r k1 1 5 dir

Since SEP 1 With this we get dt dt contains the expression t2 1 we will make the substitution t sec 8 de

Evaluate the indefinite integral 1 COS X 18 sin 8 x dx

Evaluate the indefinite integral 1 sec 30 tan 30 de C

Find the simple interest on the loan Round your answer to the nearest dollar 6750 at 4 5 for 8 months

Differentiate the function y y In 3 8x 3 8x

4 Find all solutions to the trigonometric equation tan 0 sec 0 1

Q2 Find a scalar potential field for F i j k x y z Explain why this potential field cannot be used to evaluate the line integral of F along r t ti tj f k 1 t 1 Total for Question 2 5 marks