Solving an Initial value problem by using Taylor's method

Use Taylor's method with h = 0.05 to approximate the solution, and compare it with, actual values of y.

See attached file for full problem description.

(5.3)
6. Given the initial-value problem

, ,
with exact solution :
a. Use Taylor's method with h = 0.05 to approximate the solution, and compare it with, actual values of .
b. Use the answers generated in part (a) and linear interpolation to approximate the following values of , and compare them to the actual values.
i.

Please see the attached file for the fully formatted problems.
1 Write down the Taylor polynomial of , . Use Taylor's theorem to
show that =
2. Given that , 1 < e < 3. Do the followings.
a) Write down the Taylor polynomial of and show that for and
b) Use a) to show that for al

Suppose that y(x)' : f(x,y(x)) on the interval [x0, x1] with y (x0) = y0. Assume that a unique solution y exists such that it and all of its derivatives up to and including the third order are defined and continuous on [x0, x1]. UsingTaylor's Theorem (and the Mean Value Theorem, if necessary) prove that...
See the attached f

** Please see the attached file for the complete problem description **
Solve the differential equation...using the undetermined coefficients method.
Solve the differential equation...using variation of parameters method.
Solve the initialvalueproblem...
Solve by systematic elimination....

Please see the attached file for the fully formatted problems.
Attached is a file with a three part successive approximation problem.
The following problems are to use the method of successive approximations (Picard's)
[EQUATION]
y x y fty tdt =+∫n− with a choice of initial approximation other than y0(x)=y0

1. Use the method of characteristics to solve the advection equation
du/dt=-kdu/dx-ru
subject to the initial condition u(x,0)=f(x).
2. Use the method of characteristics to solve du/dt+te^(-t^2))du/dx=usin(t) subject to the initial condition u(x,0)=e^(-x^2))
(See attachment for the above questions formatt

Consider the initialvalueproblem (IVP): y'(t) = y^2 y(0)=1
Approximate y(1) using Euler's method and step sizes of 0.25. Perform these calculations by hand (using a calculator for arithmetic is ok). What is the true value of y(1)?

Problem 27
Miller Company acquired an 80 percent interest in Taylor Company on January 1, 2009, Miller paid
$664,000 in cash to the owners of Taylor to acquire these shares. In addition, the remaining 20 percent
of Taylor shares continued to trade at a total valueof $166,000 both before and after Mil

The function of y(x) satisfies the differential equation and the initial
condition y(1)=1. Firstly solve the equation to get an exact value then use Euler's method to obtain the value of y(2). Compare this value with the analytical value and discuss how the approximate value obtained by Euler's method may be improved.
P