Differential Equations

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Differential Equations Questions and Answers

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Find F(s): L{(t - 1)u(t - 1)} where L is the laplace transformation
Find the solution r(t) of the differential equation with the given initial condition: r'(t) =
What is the frequency of blue light with a wavelength of 450 nm?
What is the frequency of blue light with a wavelength of 385 nm ?
Jon's bathtub is rectangular and its base is 12 ft^2. (a) How fast is the water level rising if Jon is filling the tub at a rate of 0.5 ft^3/min?
A rectangle has one side of 11 cm. How fast is the area of the rectangle changing at the instant when the other side is 14 cm and increasing at 2 cm per minute?
Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation: p +x^2=144 where p is measured in dollars and x is measured in units of a thousand. How fast is the quantity demanded changing when p=63, a
Consider a point moving along the curve f (x) = square root x. Find the position of the point on the curve where both coordinates of the point are changing at the same rate. If {dx} / {dt} is 2 m/sec at the point (4, f (4)), how fast is the point moving a
Find the minimum value of the function f(x) = x^4 - 18x^2 + 90, and find the values of x for which f(x) is a minimum.
You are blowing air into a spherical balloon at a rate of \frac{4 \pi}{3} \ in^3/sec. Assume the radius of your balloon is zero at time zero. Let r(t), A(t) and V(t) denote the radius, the surface area, and the volume of your balloon at time t, respective
A manufacture has been selling 1550 television sets a week at $450 each. A market survey indicates that for each $29 rebate offered to a buyer, the number of sets sold will increase by 290 per week. a. Find the function representing the demand, p(x), wher
Solve this equation for T. Show all steps. 3 + 20/(1 + e^{3(r - 37)}) + 0.8(T_{env} - T}) = 0
A company manufactures and sells x electric drills per month. The monthly cost and price-demand equations are: C(x) = 64000 + 60x; p = 220 - x/30, 0 \leq x \leq 5000 . a) Find the production level that results in the maximum revenue. b) Find the price tha
A spherical balloon is inflated at the rate of 16 \ ft^3/min. How fast is the surface area of the balloon changing at the instant the radius is 2 \ ft?
A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at a constant rate of 0.05 \ in/sec and the volume V is 128 \pi \ in^3. At what rate is the length h changing when the radius is 2.5 \ in?
As a balloon in the shape of a sphere is being inflated, volume is increasing at the rate of 4 \ in^3/sec. At what rate is the radius increasing when the radius is 1 \ in?
An inverted conical container has a diameter of 42 \ in and a depth of 15 \ in. If water is flowing out of the vertex of the container at a rate of 35 \pi \ in^3/sec, how fast is the depth of the water dropping when the height is 5 \ in?
Cars A and B leave a town at the same time. Car A heads due south at a rate of 80 \ km/hr and car B heads due west at a rate of 60 \ km/hr. How fast is the distance between the cars increasing after three hours?
A person is running around an elliptical track. The equation of the track is 4x^2+25y^2 = 200 . a. When the person is at the point (5, \ 2) , her x coordinate is increasing at 6 units per minute. Describe how her y coordinate is
A small boy standing in a flat field watches a balloon rise in the distance. The balloon leaves the ground 500 m away from the boy and rises vertically at the rate of 7 m/minute. At what rate is the angle of inclination of the boy's line of sight (in radi
Sketch the graph of each of the following, giving intercepts, asymptotes, where increasing, where decreasing, any relative maximum and relative minimum points, where concave upward, where concave downward, and any inflection points:
Two cars start moving from the same point. One travels south at 24 \ mi/h and the other travels west at 18 \ mi/h. At what rate is the distance between the cars increasing two hours later?
Two cars start moving from the same point. One travels south at 48 \ mi/h and the other travels west at 20 \ mi/h. At what rate is the distance between the cars increasing to hours later?
Use arctangents to describe the viewing angle for the sign in the figure below when the observer is x feet from the entrance to the tunnel. At what distance x is the viewing angle maximized?
Consider the function f : (0,\infty) \to R given by f(x) = x^2 + A/x^2 where A is a positive real number. Find the smallest value of A such that the function f satisfies f(x) \geq 4 for all z > 0 . (Hint: begin by finding the global minimum of f on x > 0)
The revenue derived from the production of x units of a particular commodity is R(x) =\frac {(48x-x^2)}{(x^2+48)} million dollars. What level of production results in maximum revenue? a. Maximum at x = 6 and maximum revenue is R(6) = 18 (million dollars
The daily average cost function (in RM per unit) of Electronics B is given by \overline C(x)=0.0001x^2+0.08x+40+ \frac {5000}{x}, \ \text {for} \ x0, where x stands for the number of calculators that company produces. a. Show that the production level of
A bookstore can obtain a certain gift book from the publisher at a cost of $2 per book. The bookstore has been offering the book at a price of $22 per copy and, at this price, has been selling 200 copies a month. The bookstore is planning to lower its pri
A city recreation department plans to build a rectangular playground having an area of 25,600 m^2 and surround it by a fence. How can this be done using the least amount of fencing?
You are given the price p(q) at which q units of a particular commodity can be sold and the total cost C(q) of producing the q units, where: p(q) =37-2q, \ C(q)= 3q^2+5q+75; a) Determine the level of production q where the profit P(q) is maximized. b)
A baseball team plays in a stadium that holds 48,000 spectators. When the ticket price at$8 the average attendance has been 19,000. When the price dropped to $5, the average attendance rose to 24,000. a. Find the demand function p(x), where x is the numb
Verify that y = C \sin x + \cos x is a solution to \frac{dy}{dx} = \frac{y\cos x - 1}{\sin x}. Find the unique solution that satisfies the initial value y\frac 32 \pi e = 2. What is the interval of definition for this solution?
Verify that y = \frac 1{1 + Ce^{-x}} is a solution to \frac{dy}{dx} = y - y^2. Then find the unique solution.
Find all values of the constant c such that y = c is a asolution to \frac{dy}{dx} = \frac{y^2 + 2y - 3}{3xy} .
a) verify that y = x + 4\sqrt{x + 2} is a solution to \frac{dy}{dx} = 2xy^2 b) what are all the possible intervals of definition for the solution in part a ?
Let P = (a,b) lie in the first quadrant (so a,b > 0). Find the slope of the line through P such that the triangle bounded by this line and the axes in the first quadrant has minimal area. Then, show that P is the midpoint of the hypotenuse of this triangl
A 14-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 3 feet/second, how fast is the top of the ladder moving down when the foot of the ladder is 4 feet from the wall?
Solve the differential equation y' = 4^{e^{2x}} + 6^{x^2} with the inital condition y(0) = 4.
A hot air balloon is launched from the ground and rises vertically at a rate of 10 feet per second. Barry runs toward the balloon's launch site at a rate of 4 feet per second. AT what rate is the angle of elevation (from Barry to the balloon) changing whe
A real estate office handles a 40-unit apartment complex. When the rent is $580 per month, all units are occupied. For each $25 increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of $40 per month for s
Solve the homogenous equation (14x + 3y) + (3x + y)y' = 0 with the initial condition y(2) = 7. Making the substitution u = \frac{y}{x}, you eventually get f(u) = -\ln |x| + C_1 where f(u) = and C_1 = The solution is given implicitly by F(x,y) = C, whe
Find f_{xx}(x,y), f_{xy}(x,y), f_{yx}(x,y) and f_{yy}(x,y) for the function f(x,y) = 5xe^{4xy} .
Solve for x. 6 - 2x = 5x - 9x + 4
If the population in 1950 were 2944 million and if the population in 1990 were 5401 million, what would be the true total change in population?
A closed box with a square base is to have a volume of 500 m^3. The material for the top and bottom of the box costs $4/m^2, and the material for the sides costs $1/m^2. Find the minimum possible cost for such a box.
Solve \frac{dy}{dx} + 3x^2y - 2xe^{(-x^3)} = 0, given that (x,y) = (0,1000) is part of the solution.
Find the maximum value of f(x)=x^2y^4 for x, y greater than or equal to 0 on the unit circle.
Mechanical vibrations: An automobile is found to have a natural frequency of 20 rad/s without passengers and 17.32 rad/s with passengers of mass 500kg. Find the mass and stiffness of the automobile by treating it as a single-degree-of-freedom system.
An open box is to be constructed from cardboard by cutting out squares of equal size in the corners and then folding up the sides. If the cardboard is 6 inches by 18 inches, determine the volume of the largest box which can be so constructed.
An automobile having a mass of 2000 kg deflects its suspension springs 0.02 m under static conditions. Determine the natural frequency of the automobile in the vertical direction by assuming dumping to be negligible?
A lighthouse on a rock 100 m from a straight shoreline has a rotating spotlight whose beam makes a complete rotation once per minute. At what speed, in meters per second, is the light beam traveling along the shore when it shines on a dock 500 m from the
Suppose a ladder of length 9 feet rests against a wall, and we pull the vase of the ladder horizontally away from the wall. For a given distance x that we pull the ladder horizontally, let g(x) be the height of the ladder on the wall, under the assumption
Consider the system \textbf{x}' = \begin{bmatrix} 3 & -4 \\ 2 & -3 \end{bmatrix} \textbf{x'}. Verify by hand that the vectors \textbf{x}_1 = \begin{bmatrix} 4e^{t} \\ 2e^{t} \end{bmatrix} and \textbf{x}_2 = \begin{bmatrix} e^{-t} \\ e^{-t} \end{bmat
The fuel consumption c of Micheal's car ( measured in gallons per hour ) is a function of speed ( or velocity ) v of the car ( measured in miles per hour). Since he has owned the car for several years now, Micheal has determined through experimentation th
A physical fitness room consists of a rectangular region with a semicircle at each end. If the perimeter of the room is to be a 200 ft running track, what is the radius of the semicircle that will make the rectangular region a maximum? Select one: a. 25/\
Explain why the initial value problem y' = 4x \sqrt{1 - y^2} with y(0) = 4 does not have a solution.
When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.03 cm divided by min 0.03 cm/min. At what rate is the plate's area increasing when the radius is 53 cm?
What is the period of a water wave if 4.0 complete waves pass a fixed point in 10. seconds?
An equilateral triangle has side length s which is increasing at a rate of 8 cm/s. How fast is the area a increasing when the area is 3 cm^2?
The area of a circle is increasing at a rate of 24 in^2/min. What is the rate of change of the radius at the instant that the radius is 6 in?
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