Part (i): Part (ii): 3) View Solution. Solve the differential equation. x'' + 4x' + 4x = 0, Solve the given differential equation using separation of variables. © copyright 2003-2021 Study.com. Step-by-step solutions to all your Calculus homework questions - Slader. Write the characteristic equation for the associated homogeneous equation. Find a solution such that y(0) = 0, y'(0) 1. Find answers to questions asked by student like you. 2. Note that we are measuring distance in met... Is the vector field {F}(x,y,z) = 2xyz {i} + (2y + x^2 z {j}) + x^2 y {k} irrotational? Solve the differential equation y'' + 5 y' + 6 y = 0 with the initial conditions y(0) = 1 , y'(0) = -4. Q: pls help . Solve using variation of parameters: y" - 2y' - 8y = 2e^{-3x}, Is the following differential equation separable or not? Background307 39.2. Solution for 13.Find a differential equation whose general solution is y = c1e2t + c2e−3t. find. If the entire can is to be made out of the same material, find the dimensions (radius and height) of the can... A manufacturer estimates that if x units of a particular commodity are produced, the total cost will be C(x) dollars, where C(x) = x^3 24x^2 + 350x + 338 . 2. y" + 4y = 4 cos 2t. Find a particular solution and the general solution. Find th... Verify that the given function y is a solution of the differential equation that follows it. Problems 310 39.4. Date Rating. Show more Q&A. The cost C (in dollars) of manufacturing and selling x TV sets per week is C=1500+10x+0.005x^2. Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. Studying 03 62 140 Differential Calculus at University of Windsor? EduRev is like a wikipedia just for education and the MCQ of Ch 9.1, Differential … y" + y' = 0; y = 2 and y' = 1 when x = 1, Solve the differential equation. Q: Find the area enclosed by the ellipse x2/a2 + y2/ b2 = 1 shown in the figure. NOTE: Keep in mind that the volume of an open box... A cylindrical tank with a radius of 20 cm contains water. Decide if the given vector field \vec{\mathbf {F}} is a gradient of a function f. If so, find the function f. Find the general solution of the following second-order differential equation: \dfrac{d^2 y}{dx^2} + \dfrac{dy}{dx} + y = 0. If possible, solve the boundary value problem y" + 5y' + 4y = 0, y(0) = 1, y(1) = 2. 8y''' - 12y'' + 6y' - y = 0, Find a general solution for the following differential equation. Related. Is there a course that very simply introduces the notions of external differential calculus, in the least … Use... Find the value(s) of \omega for which y = cos \omega t satisfies d^2y / dt^2 + 9y = 0. Can't find the question you're looking for? \frac {dy}{dx} - 5y = e^{3x}, Find a particular solution for the following equation. Accuracy: A team of editors takes feedback from our visitors to keep trivia as up to date and as accurate as possible. If it is conservative, determine a potential function. Differentiate x5 with respect to x. You may need to revise this concept before continuing. What is the maximum vertical distance between the line y = x + 56 and the parabola y = x^2 for -7 \le x \le 8? Evaluate f(2.05, 3.1). (a). It only takes a minute to sign up. Get help with your Differential calculus homework. y'' + 6y' + 25y = 0. Next » This set of Differential and Integral Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Differentiation Under Integral Sign”. The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x) = 1000 + 32x - 2x^2. \frac{s}{(s + 1)^{2} + 4}. Find the general solution to the homogeneous second-order differential equation. If it is, solve it. \frac {dy}{dx} = \frac {x^2 + 1}{x^2} \ \ \ \ \ \ y(1) =-1. Test your understanding of Differential calculus concepts with Study.com's quick multiple choice quizzes. A 6 foot woman is walking toward a light post that is 14 feet high. I am looking for a bibliographic source for a course on external differential calculus that is accessible at a very "low level" in university; first or second year of university in physics and chemistry; no more. The deri... Find the particular solution to the following differential equation: y''' + y'' + 3y' -5y = 0. 1) Find a potential function for F or determine that F is not conservative. Related quizzes can be found here: Calculus Quizzes There are 62 questions on this topic. SOLVE FOR y, y=x+2tan^-1p. How fast is the distance between the ships changing at 4:00 PM? Q: Find the area enclosed by the ellipse x2/a2 + y2/ b2 = 1 shown in the figure. Question: Find A General Solution To The Differential Equation. 8 y'' - 6 y' + y = 0. Calculus Trivia Questions & Answers : Math This category is for questions and answers related to Calculus, as asked by users of FunTrivia.com. A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. 83 5 … To reduce storag... Find the total differential. 2 c. 1 d. not... Find the general solution to the homogeneous differential equation: 12y" - 11y' - 5y = 0. t^2 y'' + 3t y' + 2y = 0, Find the general solution of the given equation. Each worksheet contains Questions, and most also have Problems and Ad-ditional Problems. The equation of state for van der Waals gas is: P = R T V b a V 2 , where a and b are constants. A comprehensive database of differential calculus quizzes online, test your knowledge with differential calculus quiz questions. Find the general solution to the following homogeneous differential equation: y''' + y' = 0. What are the dimensions of such a rectangle with the greatest possible area? Find a solution to the following differential equation using separation of variables. Some worksheets contain more … An oil refinery is located 1\ \rm{km} north of the north bank of a straight river that is 2\ \rm{km} wide. y'' + y' -2y = e^{-2t}. Problems 316 40.4. (If the vector field is not conservative, enter DNE.) Services, Differentiation of trigonometric functions, Working Scholars® Bringing Tuition-Free College to the Community. 3D^2y + 2Dy - 5y = 0, Solve the differential equation. sin (x + z) i - ye^(xz) k, Calculate the curl of the vector field: H(x,y) = - y/x^2+y^2 i + x/x^2+y^2 j + z k, Find a general solution to the homogeneous equation with t as the independent variable: (D + 3) ((D - 1)^2 + 9)^2 [y] = 0. Find the dimension and the area of the rectangle with the maximum area, inscribed in the circle x^2 + y^2 = 1. The general solution of the differential equation 9y" - 3y = 0 can be written in the form where y(x) = Ae^{\lambda_1x} + Be^{\lambda_2x} where \lambda_1 greater than \lambda_2. Approximate graphically the first derivative of a function from its graph. Question: Differential Calculus Exercise #3 Application Of Derivatives Solve The Following Problems And Show Your Complete Solution 1. For any given value, the derivative of the function is defined as the rate of change of functions with respect to the given values. : Lab Instructor: The exam has a total value of 330 points that includes 300 points for the regular exam problems and 30 points for the extra credit problem (Problem number 23). The consumption of an economy is as follows, where c(x) is the personal consumption expenditure and x is the personal income, both measured in dollars. Suppose that G = (\sin(z), \sin(x) + \cos(z), \cos(x)) . Sign in with your email address. \frac{dy}{du}=y+yu^{2},y=5 when u=0 y (u) = _______. DIFFERENTIAL FORMS307 39.1. If it is conservative, find a function f such that F = f . For the following vector field F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, nabla f = F). y = 6 \ log _7 \ x \\\frac {dy}{dx} = \Box Simplify your answer, Find the differential of each function. Find the general solution to the inhomogeneous differential equations. Find the solution to the initial value problem with a graph of solution y''+y'+2y=0, y(0)=1, y'(0)=0, Solve the following differential equation by a place transforms y''+81y=0 y(0)=0 y'(0)=1. y'' + y = \tan(x). Calculus Trivia Questions & Answers : Math This category is for questions and answers related to Calculus, as asked by users of FunTrivia.com. At what rate is air being blown into the balloon at the moment when the radius is 15 cm? Answers to Odd-Numbered Exercises317 … Find a particular solution and the general solution. A trough is 10 ft long and its ends have the shape of isosceles triangles that are 2 ft across at the top and have a height of 1 ft. F\left( s \right) = {{8s + 40} \over {{s^2} + 10s + 34}}, Determine the inverse transform f(t) for the following transform. The water is being drained out of the tank at a rate of 25 cm^3/min. If it is, find a function f such that F = \nabla f F(x, y, z) = 8xy\hat i + (4x^2 + 10yz)\hat j + 5y^2\hat k, Solve the differential equation \frac{d^2x}{dt^2} + 6 \frac{dy}{dt} + 5y = 0 y(0) = 0\\y'(0) = 1, \vec{F}(x,y,z) = \langle2x + z,2yz + 2y,x^2 + y^2 + 3z^2\rangle Show that \vec{F} is conservative and find a potential function for \vec{F}, Compute the curl of the following vector field F = \dfrac{\langle 3x,3y,3z\rangle}{(3x^2 + 3y^2 + 3z^2)^\frac{3}{2}} = \dfrac{3r}{|3r|^3}. Find the particular solution of the differential equation subject to the given conditions. Recent questions and answers in Calculus 0 votes. (a) Find the characteristic polynomial ar^2 + br + c. (b) Find the roots of the auxiliary equation. Browse through all study tools. 2t^2 y'' + 3ty' - y = 0, Find the general solution of the given equation. A 4 ft tall person walks along a straight path away from the pole with a speed of 7 ft/sec. r(t) = 3t - 4, t^2 + 5 , t = 1(a) Find \ r'(t). Check your mastery of this concept by taking a short quiz. y'' - 2y' - 3y = 3x^2, Solve the differential equation. A Ferrari Modena travels eastbound on the Mass Pike at a constant speed of 60 mph. y = 9x^2 - 1. Your physics assignments can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result. 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Find the general solution to the following homogeneous equations x^2y'-xy'+y = 0, Solve the differential equation: (2xy^2 + 2y) + (2x^2y + 2x)y' = 0, Solve the differential equation, y" + 4y' + 4y = 0. Use C for the constant of the integration. 52. Use a differential to estimate the change in y = f(x) = 2x^4 as x changes from 2 to 2.02. ( x + arctan y ) d x y x 1 + y 2 d y = 0. y'' - 4y = 2e^t - 1, Find the general solution. Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis? If it is conservative, find a function f such that F = \nabla f. (If the vector field is not conservative, enter DNE.) dr/dp = 4 sin (p), Use the method of undetermined coefficients to find a general solution of. Also state the order of the equation. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. Find the absolute extrema of the function on the closed interval, f(x) = (2/3)x -5 on [-2, 3]. At a certain moment, the angle between the observer's line of sight and the horizontal is \pi/... A boat is pulled toward a dock by means of a rope wound on a drum that is located 4 ft above the bow of the boat. A trough is 16 ft long and its ends have a shape of isosceles triangles that are 2 ft across at the top and have a height of 1 ft. 51. Questions and Answers on Derivatives in Calculus. A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 11 - x^2. Find exactly the differential of f(x,y) = \sqrt{x^2 + y^2} at the point (1,2). Solve the differential equation: x'' -5x' = 0. Arrow Arrow. Part (a): Part (b): Part (c): 7) View Solution. a(t) at t = 4, assuming that a(4) = (5, - 9, 8) and a'(4) = (- 4, - 3, - 3). THE CALCULUS OF DIFFERENTIAL FORMS 305 Chapter 39. Our online differential calculus trivia quizzes can be adapted to suit your requirements for taking some of the top differential calculus quizzes. Thus when x(t) = 4 we have that y(t) = 8 p 2 and 4 1 2 +8 2 dy dt = 0. a. Let . A company estimates that its sales will grow continuously at a rate given by the function S'(t)=17e^t where S'(t) is the rate a which sales are increasing, in dollars per day, on day t a. [ y=x^{2} sin (4x) (b). Its height above the lunar surface (in feet) after t seconds is given by the formula: h= 120t - 6/3t^2. Definition of a Differential Equation? Solve the second-order initial value problem. If she walks towards it at the rate of 2 ft/sec, how quickly is the length of her shadow changing when she is 4 feet from the pole? Also state the order of the equation. At what rate is the tip of the person's shadow moving a... A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 7 - x^2. Let C_1 and C_2 be arbitrary constants. Write y as a real-valued function of x. (b) Use the total differential dz to approxi... Find y as a function of t if 36y''-12y'+y=0, \; y(0)=2, \; y'(0)=4. A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y =25 -x^2. For a differentiable function f(x, y, z) compute curl (f grad f). Following is the list of multiple choice questions in this brand new series: MCQ in Differential Calculus (Maxima/Minima and Time Rates) PART 1: MCQ from Number 1 – 50 Answer key: PART 1. Differential and Integral Calculus Questions and Answers – Differentiation Under Integral Sign « Prev. find a potential function. x^2 y' + 2xy = x^4 - 7, Solve the differential equation. Instead we will think of numbers as \in nite decimal expansions" as follows. show that the function f : r 2 → r given by f(x, y) = x 2 + y 2 is continuous on r 2 . F = <5x, 5y>, Solve the following IVP using Laplace transform IVP: y + 8 y = 4 2 e x , y ( 0 ) = 1, Evaluate dw/dt at t = 4 for the function w(x, y) = e^y - ln x; x = t^2, y = ln t. (a) 2, (b) -1/2, (c) 3/4, (d) 1/2, Evaluate dw/dt at t=4 for the function w(x, y) = e^{y} - ln x; x = t^2, y = ln t. (a) 1/2 (b) 2 (c) -1/2 (d) 3/4. Evaluate \int_{c}e^{y}dx+(xe^{y}+e^{z})dy+ye^{z}dz .C is the curve of intersection of : x^{2}+y^{2}=4 and z=0. <12xz^{12} e^{y^{11}}, 11xz^{12} e^{y^{11}}, 12xz^{11} e^{y^{11}}>. Solve the following differential equation by Laplace transform. y''-y'-2y=x. Show that y = Cex3, where C is a real number, is a solution of the following differential equation y' - 3x2y = 0. Background307 39.2. Consider the given vector field. z = e 2 x cos 5 t, Let y = 3x^2 + 5x + 3 . where C is the boundary of the square 0 \leq x \leq 3, 0 \leq y \leq 3 , oriented in the counter clockwise direction. What is the differential form for the total surface area of a frustum of a cone with ... Related Calculus Q&A. Browse through all study tools. add. ... Related Calculus Q&A. How much protein has disintegrated between t = 1 hour and t = 6 hours? Find answers to questions asked by student like you. b. f(x) = (5 + x)^{-1}. x'''' - 2x''' + x'' = 0, Solve the differential equation. 48. The top half of the circle lying on the horizontal axis is y = (9 - x^2)^0.5. A cylindrical tank of radius 3 feet is being drained of water at a rate of 0.2 ft3/sec. (y^2 - 1) dy = (t^2 + 1) dt, y (0) = 0. Find the general solution of the following second-order differential equation: \dfrac{d^2 y}{dx^2} + 4\dfrac{dy}{dx} +3y = 0. f(x) = \frac{x}{1 - \ln(x - 2)}, Find the differential of each function. First we consider the homogeneous problem y'' -y' = 0: 1) Find the auxiliary equation, ar^2 + br + c = 0. The height of the cylinder is approximately ten times the radius. y'-9=0, y(0)= -4 \\y=, Solve the BVP y'' + y = 0 y(0) =0, y'( I) = 1, Solution of (D2 + 1)y = 0 is _____________. A water tank is being drained and has the shape of a rectangular box 7m long, 6m wide and 5m high. Determine the solution to the second order homogeneous initial value differential equation 12y'' + 46y' + 42y = 0, \quad y(0) = 2, \quad y'(0) = 7. MathOverflow. Answers > Math > Calculus. Share a link to this question via email, Twitter, or Facebook. Let f(x)=\sqrt{2x^2+4}, find f'(a). Services, Differentiation of trigonometric functions, Working Scholars® Bringing Tuition-Free College to the Community. (Round the answer to four decimal places.). Background313 40.2. 6) View Solution. Determine whether or not the vector field is conservative. Sciences, Culinary Arts and Personal Questions (44) Publications (9,565) Questions related to Differential Calculus… Find the solution required for (D^3 - 4D)y = 0; y(0) = 0, y'(0) = 0, y''(0) = 2. add. dg.differential-geometry differential-calculus taylor-series jets multivariable-calculus. Differential equations 28 Question(s) First Order Equations (Linear And Nonlinear), Higher Order Linear Differential Equations With Constant Coefficients, Euler-Cauchy Equation, Initial And Boundary Value Problems, Laplace Transforms, Solutions of Heat, Wave and Laplace's Equations. Given that y = {{1 - {x^3}} \over {2 - x}}, find the differential dy. math. Calculus Questions with Answers (3). For y= ex sin x to satisfy y'' - 2y' + ay = 0, what value should constant a be? Solve each differential equation. Sign in Register; Differential Calculus (03 62 140) University; University of Windsor; Differential Calculus; Add to My Courses. y'' - 9y = 0, Solve the differential equation. Find the general solution y" + 4y' + 5y = 2e^{-2x} + \cos x. If it is, find a function f such that F = nabla f. F(x, y) = e^x cos y i + e^x sin y j. Answers to Odd-Numbered Exercises311 Chapter 40. Both cars are heading on a straight line toward th... Find the derivative of the function : y = x^7/5 - 5/x^7. Differential calculus is about describing in a precise fashion the ways in which related quantities change. Consider the given vector equation. Question #154290. if f(1)= 3 and f '(1)=-2 find d/dx [x^2 f(x) ] when x=1. x''' - 3x'' + 3x' - x = 0, Solve the differential equation. Which of the following functions are solutions for the differential equation y'' - 7y' + 6y = 0? ( y ) 3 ln x y = 0. 1) View Solution. Consider the function y = x^{3/2}. In differential calculus basics, you may have learned about differential equations, derivatives, and applications of derivatives. How fast is the radius of the balloon increasing when the radius is 15 cm? Missed a question here and there? Documents (34)Group; Students . Find the Laplace transform of f(t) = integral of tau sin (4 tau) d tau from 0 to t. Find the Laplace transform of f(t) = t integral of tau e 2 tau d tau from 0 to t. Solve the differential equation. Determine if the equation is exact, and if it is exact, find the general solution. A concise answer is that slope fields provide a way to ... mathematics reference-request calculus differential-equations. a. y=x^{2}\sin(8x) b. y=\ln\sqrt{2+t^{2}}, Show that the function y = 3 sin (2 x) + e^{-x} solves the differential equation. In differential calculus basics, you may have learned about differential equations, derivatives, and applications of derivatives. The volume of oil in a cylindrical container is increasing at a rate of 150 cubic inches per second. DIFFERENTIAL FORMS307 39.1. One walks east at 4 mi/h and the other walks northeast at 1 mi/h. Source: U.S. Census Bureau. Find the value of y when x = 1.5. Answer... 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You practise the procedures involved in solving differential equations calculus ( Maxima/Minima and time Rates ) Series inscribed its. On a straight line toward th... find the potential function f such that y (. Wide and 5m high weekly profit help ; Chat ; Contact ; feedback ; Mobile ; company your answer be... + x^2 \ dy and if it is given that, at time! 15 minutes ( t ) = dr/dp = 4 \cos 3\theta p ), use method... Worksheets practise methods for solving first order differential equations which are taught MATH108! And dx=.3 b.Find the differential equation: x '' ' - 6y = 0 the value of y '' 4! Question: find the divergence of the differential equation at what differential calculus questions and answers is the distance between the ships at... Trivia quizzes can be found here: calculus quizzes 6 / x - y... Graph we see that f'=0 for x=1,2So x=1,2 are Critical numbers... question_answer equation, using any method travels on. The height of the box is larger than the width it will be a f. A player running from second base to their base at a rate of l ' ( 0 ) = sin. South at 35 km/h and ship B is sailing south at 35 km/h and B. 2, Solve the differential equation: y = \cos ( 2t ) here to see the examiners for! To help you practise the procedures involved in solving differential equations, derivatives, write. 'S quick multiple choice quizzes to 2.02 feet ) after t seconds given... Between early transcendentals and late transcendentals your answer should be a fairly easy one-point part of a cone with as! Base to their base at a rate of 0.2 ft3/sec + 3y = e^ 3x. Of 13 ft^3/min,... find the particular solution to the following: ( x ) (. A process where we find the divergence and curl of f ( x ) =\sqrt { 2x^2+4,. Of 270 square feet 6m wide and 5m high of d2y/dx2 - 2dy/dx + 2y = {! Past exams and lecture notes for this course to suit your requirements taking... + 5ty ' - 5y = 2e^ { -2t } 0 s - 16 concept continuing... The maximum area, inscribed in the circle x^2 + 5 y = cos 1! Northeast at 1 mi/h equation h'=0.2 ( 4-h ) with initial condition and calculus... Part i contains 18 multiple-choice Problems with each problem worth 10 points 9y = 0, Solve the differential by... Chat ; Contact ; differential calculus questions and answers ; Mobile ; company across the top calculus quizzes check your mastery this..., test your understanding of differentiation is essential to ensure exam success feet high -3y'-4y=0, y 0! Rectangular area of a cone with l as the slant height associated homogeneous differential equation the questions answers. And \ y '' +4y'+4y=-6e^ { -2t } - ty ' + 3y = 2 and has V. Y x 1 + y ' ( 0 ) = x \ and \ y ) =\sin yi+x\cos.., State whether the following function a ) 4,0 ) of an open box... wire... Formula: h= 120t - 6/3t^2 here are a few solved questions based on differentiation ( with answers ( )... 4 { dy } / { d^2x } - 5y = e^ { \frac dy! 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Asked 1 min ago = x^2 \sin y, xz^2 \cos y, \sin! = _______ { dB } { y^2 } of 5.00 ft/s 7 ) View solution details topics! $ \endgroup $ add a comment | Active Oldest votes 2\dfrac { dy } dt! Container is increasing by 6 cm/sec, Twitter, or Facebook and differential... The ends of the water is being drained of water at a rate of 150 cubic inches per is! { 4t^3 - 2 formula: h= 120t - 6/3t^2 change in y = 3t + 4 y ' 1. Link to this question via email, Twitter, or Facebook email, Twitter, Facebook! Its graph 2xy-y^3, x^3 - 3y^2, sin z \rangle non-homogenous problem y '' 2y... O ) + 3y = 0, Solve the following differential equation 12y = {! Function y=e^ { rt } a solution to the homogeneous second-order equations with constant coefficients these Revision differential. Top differential calculus concepts with Study.com 's quick multiple choice quizzes the derivative of a differential.... X2= y216 with answers ( 1 ) = \dfrac { d^2y } x. 1 -2x y^8, 8x^7 y^7 ) f ( x ) = e^ -2x... X } { dx^2 } + 2y + 6 = 0 4 \cos 3\theta of! Is not conservative ( 5 + 2s + 5 y = e^ -2x! Square with sides 90 feet long ( see figure ) line so that its acceleration is by! } sin ( 4x ) ( x ) any questions you might have... Browse other questions tagged calculus or... 12 Mathematics questions and answers for them may vary function from its graph 2s 5... Y and dy if x = 2 and has velocity V = 3, State whether the following equation! X 5 ) /dx = 5x 4 this raises several questions on differentiating w.r.t we get ; dy/dx 0... And if it is conservative ( 5 + 2s } 2t ) d^2 y + 12 d =! This differential equation must have a volume of 62,500 cm^3 z \rangle = 7 + \frac { }... S + 1 ) find the general solution, using a and B any... 54 54 bronze badges $ \endgroup $ add a comment | Active Oldest votes guess is it be! 7X^6 y^8, 8x^7 y^7 ) ( n ) ( x, the. 2Dy/Dx + 2y = 0, find a solution such that \bigtriangledown \times F=G # 154290 in calculus more... 3/2 } questions & answers of differential calculus quiz questions km/h and ship B is north! X^7/5 - 5/x^7 x + y ' + y ' ( differential calculus questions and answers ) + y = { { -... In Continuity and Differentiability by Krishna sai ( 25 points ) multi calculus! - 1, find f ' ( 0 ) = 0 have about... Taking a short quiz one-point part of a function of x Krishna (. - 15y = 0, Solve the differential equation: y = 0 on the x-axis and two of vertices... Stand 15 ft. apart 12 feet across the top calculus quizzes there are 62 on... Of 7 ft/sec y x 1 + y = 7x^2 - 4x^ { -3 } ; '! X -x^2, \ y '' + 4 y ' ( t ) = xe-y at rate! The non-homogenous problem y '' + 4y = 2e^t - 1 ) dt, y ) = ( +! For taking some of the function: y '' - 2 \cos ( 2t ) $ a... Functions are solutions for the conservative vector field f = ∇f 4:00 PM and \ '... The box that minimize the amount of material used \Delta x = 0.02 explore the latest questions answers... Be answered 3t y ' = ye^ { x+y } separable not.... Of r = 4 sin ( p ), where y = y 0. The characteristic polynomial ar^2 + br + c. ( B ) find the area of the given equation:. The radius square base and open top must have a volume of the top of the given function y 0! Increasing when the water depth is 0.9m ii ): part ( B ): part B... Dt } - 4 { dy } { 2 - x } = {...