Valerie is making a corral with 700 feet of fencing. She is…
Valerie is making a corral with 700 feet of fencing. She is making a rectangular corral and then using two lines of fencing to divide it into three equally sized rectangular pens. (See image). What dimensions should the corral have in order to maximize the total area of the Corrals?
Read DetailsExistence-Uniqueness Theorem: If f(x, y) and df/dy are conti…
Existence-Uniqueness Theorem: If f(x, y) and df/dy are continuous on a rectangle R in the xy-plane containing the initial condition y(x0)=y0, then the initial value problem y’=f(x,y), y(x0)=y0 has a unique solution in R. 6pts Determine whether the Existence-Uniqueness Theorem can be used to determine if the initial value problem: y’ = 1/x + y1/3, (1,1) has a unique solution. Please indicate the largest possible rectangle R from the Theorem. 21pts First order ODEs: Solve the following. Provide solutions in explicit form if possible. Theorem: M(x,y) dx + N(x,y) dy = 0 is an exact equation if dM/dy = dN/dx. a. (y4 + 1)cos x dx – y3 dy = 0 b. (12x – y)dx – 3x dy = 0 c. (x3 + y/x)dx + (y2 + ln x) dy = 0 8pts Homogeneous ODE: Solve y iv + 5y ‘’ – 36y = 0. 10pts Nonhomogeneous ODEs: Solve the following with either undetermined coefficients or variation of parameters to solve 3y ‘’ – y’ – 2y = 4x + 1, y(0) = 1 and y’(0) = 0 10pts Systems: Solve the following. x1’ = 2×1 – 4×2 x2’ = 2×1 – 2×2 15pts Solve the initial value problem for y(t) using the method of Laplace transforms. y ’’ + 4y’ + 3y = 1 y(0)=0, y’(0) = 0 Taylor polynomial about 0: pn(x) = f(0) + f’(0)x + f ‘’(0)/2! x2 + f ‘’’(0)/3! x3 + … + f (n)(0)/n! xn 15pts Determine the first three nonzero terms in the Taylor polynomial approximations for the given initial value problem y ’’ – 2y’ + y = 0; y(0)=0, y’(0) = 1 Theorem: Consider the differential equation A(x) y” + B(x) y’ + C(x) y = 0. If the functions p(x) = B(x)/A(x) and q(x) = C(x)/A(x) are analytic at x =0, then the general solution is produced by the power series centered at x=0: y(x) = a0 + a1x + a2 x2 + a3 x3 + … 15pts Determine the first four nonzero terms in the power series expansion about x=0 for a general solution in the given ODE y ’’ + xy’ + y = 0
Read DetailsAcademic Misconduct Policy: I understand that no outside r…
Academic Misconduct Policy: I understand that no outside resources (ex. phones, notes, etc.) are allowed on this exam, or on any exams in this course. I also understand that I am not allowed to communicate with anyone during the exam. This includes reading the questions out loud. Use of outside resources or communicating with others during the exam will result in a zero for the exam, and will be reported to the Academic Dean for possible further disciplinary action. I understand that in order to receive credit for the exam, the webcam must show myself, my monitor, and my desk or table surface. I also must be focused on the computer screen throughout the exam. Intentionally pointing the webcam away, or looking away from the computer will result in a zero for the exam. Please enter your name below:
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