Elementary Differential Equations
10th Edition
ISBN: 9780470458327
Author: William E. Boyce, Richard C. DiPrima
Publisher: Wiley, John & Sons, Incorporated
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Chapter 1.3, Problem 5P
To determine
The order of the given
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4. The runway at the Piarco International airport has an equation of -3(x-2y) = 6. If
the Priority Bus Route passes through the geometric coordinate (1,-9) and is
perpendicular to the runway at the Piarco International airport. Determine the
following:
a. State two geometric coordinates which the runway at the Piarco International
airport passes through.
b. Derive the equation of the Priority Bus Route.
[2 marks]
[6 marks]
Q4*) Find the extremals y, z of the the functional
I =
1
(2yz - 2x² + y²²
12
-
212) dx,
with y(0) = 0, y(1) = 1, z(0) = 0, ≈(1) = 0.
Solve the following initial value problem over the interval from t= 0 to 2
where y(0)=1.
dy
yt² - 1.1y
dt
Using Euler's method with h=0.5 and 0.25.
Chapter 1 Solutions
Elementary Differential Equations
Ch. 1.1 - In each of Problems 1 through 6, draw a direction...Ch. 1.1 - In each of Problems 1 through 6, draw a direction...Ch. 1.1 - In each of Problems 1 through 6, draw a direction...Ch. 1.1 - In each of Problems 1 through 6, draw a direction...Ch. 1.1 - In each of Problems 1 through 6, draw a direction...Ch. 1.1 - In each of Problems 1 through 6, draw a direction...Ch. 1.1 - In each of Problems 7 through 10, write down a...Ch. 1.1 - Prob. 8PCh. 1.1 - Prob. 9PCh. 1.1 - Prob. 10P
Ch. 1.1 - In each of Problems 11 through 14, draw a...Ch. 1.1 - Prob. 12PCh. 1.1 - Prob. 13PCh. 1.1 - Prob. 14PCh. 1.1 - Consider the following list of differential...Ch. 1.1 - Prob. 16PCh. 1.1 - Prob. 17PCh. 1.1 - Consider the following list of differential...Ch. 1.1 - Consider the following list of differential...Ch. 1.1 - Prob. 20PCh. 1.1 - Prob. 21PCh. 1.1 - A spherical raindrop evaporates at a rate...Ch. 1.1 - Newton’s law of cooling states that the...Ch. 1.1 - A certain drug is being administered intravenously...Ch. 1.1 - Prob. 25PCh. 1.1 - Prob. 26PCh. 1.1 - Prob. 27PCh. 1.1 - In each of Problems 26 through 33, draw a...Ch. 1.1 - Prob. 29PCh. 1.1 - In each of Problems 26 through 33, draw a...Ch. 1.1 - In each of Problems 26 through 33, draw a...Ch. 1.2 - Solve each of the following initial value problems...Ch. 1.2 - Prob. 2PCh. 1.2 - Prob. 3PCh. 1.2 - Prob. 4PCh. 1.2 - Prob. 5PCh. 1.2 - Prob. 6PCh. 1.2 - The field mouse population in Example 1 satisfies...Ch. 1.2 - Consider a population p of field mice that grows...Ch. 1.2 - The falling object in Example 2 satisfies the...Ch. 1.2 - Prob. 10PCh. 1.2 - Prob. 11PCh. 1.2 - A radioactive material, such as the isotope...Ch. 1.2 - Prob. 13PCh. 1.2 - Prob. 14PCh. 1.2 - Prob. 15PCh. 1.2 - Prob. 16PCh. 1.2 - Consider an electric circuit containing a...Ch. 1.2 - Prob. 18PCh. 1.2 - Prob. 19PCh. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - Prob. 3PCh. 1.3 - Prob. 4PCh. 1.3 - Prob. 5PCh. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - Prob. 7PCh. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 7 through 14, verify that each...Ch. 1.3 - In each of Problems 15 through 18, determine the...Ch. 1.3 - In each of Problems 15 through 18, determine the...Ch. 1.3 - In each of Problems 15 through 18, determine the...Ch. 1.3 - In each of Problems 15 through 18, determine the...Ch. 1.3 - In each of Problems 19 and 20, determine the...Ch. 1.3 - In each of Problems 19 and 20, determine the...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems 25 through 28, verify that...Ch. 1.3 - Prob. 26PCh. 1.3 - Prob. 27PCh. 1.3 - Prob. 28PCh. 1.3 - Prob. 30PCh. 1.3 - Prob. 31P
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- Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral I = = F(x, y, y″) dx. Hence write down a first integral of the Euler-Lagrange equation for the integral I 1 = √(xy ² + x³y²) dx. Find the general solution of this ordinary differential equation, seeking first the complementary function and then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try functions proportional to log x.)arrow_forwardQ2*) In question P3 we showed that a minimal surface of revolution is given by revolution (about the x-axis) of the catenary, with equation y = C cosh ((x – B)/C). - (a) Suppose, without loss of generality, that the catenary passes through the initial point P = (x1,y1) = (0, 1). First deduce an expression for the one-parameter family of catenaries passing through point P. Next calculate the value of x at which y takes its minimum value. By using the inequality cosh > √2 (you might like to think about how to prove this), show that there are points Q for which it is impossible to find a catenary passing through both P and Q. In particular, show that it is impossible to find a catenary joining the points (0, 1) and (2, 1). (b) A minimal surface of revolution can be realised experimentally by soap films attached to circular wire frames (see this link and this link for examples). The physical reason for this is that the surface tension, which is proportional to the area, is being minimised.…arrow_forwardQ3*) Consider the integral I Yn, Y₁, Y2, . . ., Y'n) dã, [F(x, Y 1, Y2, · · Yng) = - where y1, 2, ...y are dependent variables, dependent on x. If F is not explicitly dependent on x, deduce the equivalent of the Beltrami identity. Optional: Give an example of a function F(y1, Y2, Y₁, y2), and write down the Euler-Lagrange equations and Beltrami Identity for your example. Does having this Beltrami Identity help solve the problem?arrow_forward
- Solve the following problem over the interval from x=0 to 1 using a step size of 0.25 where y(0)= 1. dy = dt (1+4t)√√y (a) Euler's method. (b) Heun's methodarrow_forwardNo chatgpt pls will upvotearrow_forwardUse Euler method to solve y' = y + x, h=0.2, y(0)=0, 0 ≤ x ≤ 1. Also, find the exact solution and the absolute error.arrow_forward
- Evaluate = f J dx by using Simpson's rule, 2n=10. 2arrow_forwardUse Euler and Heun methods to solve y' = 2y-x, h=0.1, y(0)=0, compute y₁ y5, calculate the Abs_Error.arrow_forwardUse Heun's method to numerically integrate dy dx = -2x3 +12x² - 20x+8.5 from x=0 to x=4 with a step size of 0.5. The initial condition at x=0 is y=1. Recall that the exact solution is given by y = -0.5x + 4x³- 10x² + 8.5x+1arrow_forward
- B: Study the stability of critical points of ODES: *+(x²-2x²-1)x+x=0 and draw the phase portrait.arrow_forwardB: Study the stability of critical points of ODEs: -2x²+x²+x-2=0 and draw the phase portrait.arrow_forward2/ Draw the phase portrait and determine the stability of critical point: ✗ 00 +2X°-x²+1=0arrow_forward
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