Differential equations Questions and Answers

Determine whether the series is convergent or divergent 1 57 87 n 1 O convergent O divergent If it is convergent find its sum If the quantity diverges enter DIVERGES

An isotope of strontium Sr has a half life of 25 years Find the mass of Sr that remains from a sample of 30 mg after years Om 30e t m 30e t Om 30e t Om 30et Question 10 Continuing the isotope of strontium problem how long it would take for the mass to decay to 10 mg O about 40 years O about 30 years O about 50 years 10 pts

Solve the differential equation dy y 6z y Please show your work to receive full credit

Given that all members of the family y Ce are solutions of the differential equation y 3x y find a solution that satisfies the initial condition y 0 15 O y 15e Oy 30e O y 45e O y 5e

1 point Let y be the solution of the initial value problem The maximum value of y is y y sin 2x y 0 0 y 0 0

a Find a particular solution to the nonhomogeneous differential equation y 6y 16y ex b Find the most general solution to the associated homogeneous differential equation Use c and c in your answer to denote arbitrary constants and enter them as c1 and c2 Yh help formulas y help formulas c Find the most general solution to the original nonhomogeneous differential equation Use c and c in your answer to denote arbitrary constants help formulas

Find a Cartesian equation for the curve r 3 sec 0 Identify the curve O hyperbola O ellipse O lima on O line O circle

2 A tank with a capacity of 500 liters originally contains 200 liters of water with 100 kilograms of salt in the solution Water containing 1 kg of salt per liter is entering at a rate of 3 liter min and the mixture is allowed to flow out of the tank at a rate of 2liter min Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow Find the concentration in kilograms per liter of salt in the tank at the instant it starts to overflow

Check if the equation is exact if it is solve it If there is an initial value find the particular solution satisifying the initial value 1 3x 2 y 2 4xy y 2y 2 2xy 3

1 An object stretches a spring 4cm in equilibrium Find its displacement for t 0 if it is initially displaced 36 cm above equilibrium and given a downward velocity of 25 cm s

Find the general solution and if an initial value is given solve the IVP 1 y 5y 6y 18x 2 18x 22

Solve the differential equation If there is an initia value find the particular solution satisfying the initial value 1 xy 2y 4x 2 y 1 4

Consider the following graph 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 equatico and justify your answer For points in the first quadrant the differential equation y 1 xy Select graphed Select in the first quadrant and Select For points in the first quadrant the differential equation y 2xy Select graphed Select in the first quadrant and Select For points in the first quadrant the differential equation y 1 2xy Select graphed Select in the first quadrant and Select To summarize the graph is the solution to which of the three differential equations O A y 1 xy OB y 2xy O c y 1 2xy For points in the second quadrant this differential equation Select v For points in the second quadrant this differential equation Select MY NOTES Since the in the second quadrant this differential equation Select be the correct equati V Since the fa in the second quadrant this differential equation Select be the correct equatie v Since the in the second quadrant this differential equation Select be the correct equation For points in the second quadrant this differential equation Select ASK

Find the average value of the function f p 4 0 p over the solid ball p 9 The average value is 27 4 Simplify your answer Type an exact answer using x as needed

Consider the following graph YA B y 2xy The function in the graph is a solution of one of the following differential equations A y 1 xy 0 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 in the first quadrant and Select Select For points in the first quadrant the differential equation y 2xy Select in the first quadrant and Select Select For points in the first quadrant the differential equation y 1 2xy Select in the first quadrant and Select Select To summarize the graph is the solution to which of the three differential equations OA y 1 xy OB y 2xy Oc y 1 2xy Need Help Read It V Since the function grap For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation V Since the function graph 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 in the second quadrant this differential equation Select be the correct equation Since the function gra

A graphing calculator is recommended A hawk flying at 19 m s at an altitude of 228 m accidentally drops its prey The parabolic trajectory of the falling prey is described by the equation x 57 y 228 until it hits the ground where y is its height above the ground and x is its horizontal distance traveled in meters Calculate the distance traveled in m by the prey from the time it is dropped until the time it hits the ground Express your answer correct to the n tenth of a meter 185265 004 Xm

Evaluate the integral 0 2 cos 0 de

Solve for x 3 cos x 1 cos x over the interval 0 2 If needed give your answer s as an approximation rounded to the nearest 2 decimal places If there is more than one answer separate your answers using a comma with no spaces between symbols Ex 5 34 3 20

Solve for x cos x cos 2x 0 X 0 2

Use the divergence theorem to find the outward flux of F across the boundary of the region D F 2y x i z y j 2y 2x k D The cube bounded by the planes x 3 y 3 and z 3 The outward flux is Type an exact answor

1 point Estimate f x g x dx if f x x and g x has the values given in the following table 3 o f x g x dx 26 169 x 00 5 1 1 5 2 2 5 3 g x 11 11 62 63 64 14 3

We now have u t tan t C Since u 0 9 we can substitute into the equation and solve for C 81 Part 5 of 5 Noting that u 0 9 is negative then when we solve for u we must get u Submit Skip you cannot come back 81

Tutorial Exercise Find the solution of the differential equation that satisfies the given initial condition du 2t sec t 2u Part 1 of 5 Separating du dt u 0 9 2t sec t 2u gives us 2u du dt

10 Points DETAILS Find the solution of the differential equation that satisfies the given initial condition dy 4 y 0 5 dx y Submit Answer 5 10 Points SCALCCCH4 DETAILS SCALCCCH4 7 3 013 MI SA This question has several parts that must be completed sequentially If you skip a par able to come back to the skipped part Tutorial Exercise Find the solution of the differential equation that satisfies the given initial condit 2t sec t 0 9

du To separate 2 2u t tu factor the right hand side to get dt Part 2 of 6 Seperating the expression in u from the expression in t we get du 2 t dt du dt 2 t 1 u 1 u

An industrial fan blade rotates 80 times every three minutes If the blade s diameter is 6 ft find the linear and angular velocities of the tip of the blade rounded to the nearest whole number

The radius of a wheel rolling on the ground is 78 centimeters If the wheel rotates through an angle of 45 how many centimeters does it move Express your answer in terms of and then round to two decimal places The distance that the wheel moves is centimeters Type an exact answer using as needed Use integers or fractions for any numbers in the expression The distance that the wheel moves is about centimeters Round to two decimal places as needed

If tan 0 O 3 5 3 5 O O 4 5 1 2 m 334 and lies in the fourth quadrant then cos 0

Solve for the exact solutions in the interval 0 2 List your answers separated by a comma if it has no real solutions enter DNE 3 sin 2 2 3 sin 3

3 Find an equation for the following polynomial 34 16 3 2 1 0

We have f x Part 2 of 3 ae 2 be 1 Submit Skip you cannot come back ac be and f x ae Substituting and combining like terms we get 4f x 2f x 2f x 4 be ex 2 ae be X

a What can you say about a solution of the equation y 1 3 y2 just by looking at the differential equation O The function y must be strictly decreasing on any interval on which it is defined The function y must be increasing or equal to 0 on any interval on which it is defined O The function y must be strictly increasing on any interval on which it is defined The function y must be equal to 0 on any interval on which it is defined The function y must be decreasing or equal to 0 on any interval on which it is defined b Verify that all members of the family y 3 x C are solutions of the equation in part a x c y X C LHS y Y x c X C d Find a solution of the initial value problem y 1 3 y y 0 0 1 1 1 1 R RHS c Can you think of a solution of the differential equation y 1 3 y that is not a member of the family in part b Oy x is a solution of y 1 3 y2 that is not a member of the family in part b B O Every solution of y 1 3 y is a member of the family in part b O y 0 is a solution of y 1 3 y2 that is not a member of the family in part b O y x is a solution of y 1 3 y that is not a member of the family in part b O y 3 is a solution of y 1 3 y2 that is not a member of the family in part b

Use Cramer s rule to solve the system 9w 8x 10y z 37 w x y z 6 w x y z 6 4w 9x 10y z 70 Find the determinants Let A be the coefficient matrix of the given system of equati A Pw D W Dx

X Find an equation of the curve that passes through the point 0 6 and whose slope at x y is 2 x y 6 X Need Help Read It

Find the area of the region enclosed by one loop of the curve sin 40 r

Given that AABC has mA 63 b 9 and c 11 find the remaining side length a and angles B and C rounded to the nearest tenth Provide your answer below A 63 9 11 C a B

A large room has a temperature of 70 degrees Fahrenheit When a cup of tea is brought into the room the tea has a 70 represents the situation where t is time in temperature of 120 degrees Fahrenheit The function f t minutes C is a constant and k is a constant Cel kt After 3 minutes the cup of tea has a temperature of 100 degrees What will be the temperature of the tea in degrees Fahrenheit after 5 minutes Round your answer to the nearest tenth and do not include units

Tutorial Exercise Find the Maclaurin series for the function f x cos 9x Step 1 To find Maclaurin series for the function f x cos 9x recall the power series for the elementary function cos x COS X 00 1 x2n 2n

8 Taylor s theorem generates a power series from a function using the formula f x f n c x c n n 0 True O False

Find the domain and range of the given function and identify its level curves Sketch a typical level 2 f x y qx y curve

Evaluate fzyldz y dz 2 dy where C is the rectangle with vertices 0 0 4 0 4 1 and 0 1

1 point Determine the bifurcation value s for the one parameter family k help numbers dy dt Determine which differential equation corresponds to each phase line You should be able to state briefly how you know your choices are correct 1 K equal to the bifurcation value 2 k larger than the bifurcation value 3 K smaller than the bifurcation value y k A B

Find the absolute maximum value of y 36 x on the interval 6 6 Select the correct answer 0 1 05 6 7

1 point Consider the initial value problem y 3 2y y 2y 0 y 0 1 y 0 2 y 0 0 Suppose we know that y t e y2 t y3 t e21 are three linearly independentt solutions Find a particular solution satisfying the given initial conditions y t help formulas

1 point Find the general solution to y y 7y 7y 0 In your answer use c c and c3 to denote arbitrary constants and x the independent variable Enter c as c1 c as c2 and c3 as c3 help equations

1 point Find the general solution to 3y 108y 0 Give your answer as y In your answer use c and c to denote arbitrary constants and x the independent variable Enter c as c1 and c as c2 help equations

y c1 e 0 x c2 e 3 x y 1 33 e 0 x 0 33 e 3x y c e x c e 3x y 1 33e x 0 33e 3x Find the particular solution that satisfies y 0 1 and y 0 1 y 1 33e 0x 0 33e 3x help equations incorrect incorrect least one of the answers above is NOT correct point a Find the general solution to y 3y 0 Give your answer as y In your answer use c and c to denote arbitrary constants and x me independent variable Enter c as c1 and as c2 y c1e 0x c2e 3x help equations

point Find the general solution to y 4y 29y 0 Give your answer as y In your answer use c and to denote arbitrary constants nd x the independent variable Enter c as c1 and c as c2 help equations

1 point Are the functions f g and h given below linearly independent f x e x g x xex h x x e x If they are independent enter all zeroes If they are not linearly independent find a nontrivial solution to the equation below Be sure you can justify your answer e4x xex x e4x 0

1 point a Find the general solution to y 10y 25y 0 Enter your answer as y In your answer use c and c to denote arbitrary constants and x the independent variable Enter c as c1 and c as c2 help equations b Find the solution that satisfies the initial conditions y 0 4 and y 0 0 help equations