Question

4. For the initial-value problem y′(t) = 3 + t − y(t), y(0) = 1:

(i) Find approximate values of the solution at t = 0.1, 0.2, 0.3, and 0.4 using the Euler

method with h = 0.1.

(ii) Repeat part (i) with h = 0.05. Compare the results with those found in (i).

(iii) Find the exact solution y = y(t) and evaluate y(t) at t = 0.1, 0.2, 0.3, and 0.4. Compare these values with the results of (i) and (ii).

Answer #1

Use Euler's method to approximate y(0.2), where
y(x) is the solution of the initial-value
problem
y'' − 4y' + 4y = 0, y(0) = −3, y'(0) =
1.
Use
h = 0.1.
Find the analytic solution of the problem, and compare the
actual value of y(0.2) with y2. (Round
your answers to four decimal places.)
y(0.2)
≈
(Euler approximation)
y(0.2)
=
-2.3869
(exact value)
I'm looking for the Euler approximation number, thanks.

6. Consider the initial value problem
y' = ty^2 + y, y(0) = 0.25,
with (exact) solution y(t).
(a) Verify that the solution of the initial value problem is
y(t) = 1/(3e^(-t) − t + 1)
and evaluate y(1) to at least four decimal places.
(b) Use Euler’s method to approximate y(1), using a step size of
h = 0.5, and evaluate the difference between y(1) and the Euler’s
method approximation.
(c) Use MATLAB to implement Euler’s method with each...

y’ – y = 2x -1 y(0) = 1 , 0 ≤ x ≤ 0.2
Use the Euler method to solve the following initial value
problem
(a) Check whether the function y = 2 ex -2x-
1 is the analytical solution ;
(b) Find the errors by comparing the exact values you’re your
numerical results (h = 0.05 and h = 0.1)
and Discuss the issue of numerical stability.

(a) Use Euler's method with each of the following step sizes to
estimate the value of y(0.4), where y is the
solution of the initial-value problem
y' = y,
y(0) = 9.
(i) h = 0.4
y(0.4) =
(ii) h = 0.2
y(0.4) =
(iii) h = 0.1
y(0.4) =
(c) The error in Euler's method is the difference between the
exact value and the approximate value. Find the errors made in part
(a) in using Euler's method to estimate the true value...

Use Euler's method to approximate y(1.2), where
y(x) is the solution of the initial-value
problem
x2y'' − 2xy' + 2y = 0, y(1) =
9, y'(1) = 9,
where
x > 0.
Use
h = 0.1.
Find the analytic solution of the problem, and compare the
actual value of y(1.2) with y2. (Round
your answers to four decimal places.)
y(1.2)
≈
(Euler approximation)
y(1.2)
=
(exact value)

Consider the following initial value problem:
dy/dt = -3 - 2 *
t2, y(0) = 2
With the use of Euler's method, we would like to find an
approximate solution with the step size h = 0.05 .
What is the approximation of y
(0.2)?

Given the initial value problem:
y'=6√(t+y), y(0)=1
Use Euler's method with step size h = 0.1 to estimate:
y(0.1) =
y(0.2) =

The Duffing equation y'' + y + y 3 = 0 is a model for vibrations
of a mass attached to nonlinear spring. Use the vector Euler method
with h = 0.2 to approximate the solution at t = 0.6 when the
initial conditions are: y(0) = 0, y'(0) = 1.

Find the solution of the initial value problem
y′′+4y=t^2+4^(et), y(0)=0, y′(0)=3.

Solve the initial value problem below for the Cauchy-Euler
equation
t^2y"(t)+10ty'(t)+20y(t)=0, y(1)=0, y'(1)=2
y(t)=

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