Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 1, Problem 6A
Reduce to halves.
a.
b.
c.
d.
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Students have asked these similar questions
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.
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.)
Chapter 1 Solutions
Mathematics For Machine Technology
Ch. 1 - Write the fractional part that each length, A...Ch. 1 - A welded support base is cut into four pieces as...Ch. 1 - The circle in Figure 1-5 is divided into equal...Ch. 1 - The circle in Figure 1-5 is divided into equal...Ch. 1 - Reduce to halves. a. 48 b. 918 c. 100200 d. 121242Ch. 1 - Reduce to halves. a. 2510 b. 1812 c. 12636 d....Ch. 1 - Reduce numbers to lowest terms in Exercises 7 and...Ch. 1 - Reduce numbers to lowest terms in Exercises 7 and...Ch. 1 - Express the fractions in Exercises 9 and 10 as...Ch. 1 - Express the fractions in Exercises 9 and 10 as...
Ch. 1 - In Exercises 11 and 12, express the given...Ch. 1 - In Exercises 11 and 12, express the given...Ch. 1 - Express the mixed numbers in Exercises 13 and 14...Ch. 1 - Express the mixed numbers in Exercises 13 and 14...Ch. 1 - Express the improper fractions in Exercises 15 and...Ch. 1 - Express the improper fractions in Exercises 15 and...Ch. 1 - Express the mixed numbers in Exercises 17 and 18...Ch. 1 - Express the mixed numbers in Exercises 17 and 18...Ch. 1 - Sketch and redimension the plate shown in Figure...
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- Q2*) 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_forwardSolve 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_forward
- Use 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_forwardB: Study the stability of critical points of ODES: *+(x²-2x²-1)x+x=0 and draw the phase portrait.arrow_forward
- B: 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_forwardstudy the stability of critical point of oDES: 2 200+ (x² - 2x² - 1) + x=0 and draw the phase portrait.arrow_forward
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