Question

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 step size

i. h = 0.25, ii. h =, 0.125, iii. h = 0.0625,

all for 0 ≤ t ≤ 1 and plot the exact solution from part (a) and the three Euler’s method results for this part in a single figure. Include a title and a legend. Save the figure as hw1q6c.fig. Record the numerical values of each of the three Euler’s method approximations of y(1), to be used in part (d). Use the function euler provided in the file euler.m.

(d) Let yn[h] denote the result of Euler’s method after n steps of size h. Calculate (by hand, or use MATLAB and write down the results) the errors

i. E4[0.25] = y(1) − y4[0.25], ii. E8[0.125] = y(1) − y8[0.125], iii. E16[0.0625] = y(1) − y16[0.0625].

Notice that the sign of the error En[h] (nh = 1) tells us if the Euler’s method approximation is less than, or greater than, the exact value y(1). From your calculations i.–iii., which Euler’s method approximation of y(1) from part (c) is determined to be the most accurate (has the least absolute value |En[h]|)? Guess what the error E32[0.03125] = y(1) − y32[0.03125] would be, approximately.

Answer #1

**6.) (a) Solve the Differential equation.**

**Step 1.) First, solve the Differential Equation the put
the value of t = 0 in the following finding.**

Divide the above equation with y^{2}.

Substitute,

On differentiate we get,

Substitute the equation (2) in equation (1).

The above equation become,

Now, the above equation is a linear first-order ordinary differential equation.

It can be solved by using the integrating factor.

Equation (3) is in the form of

P(t) = 1, Q(t) = -t

**Find the Integrating factor.**

P(t) = 1

**By using Integrating factor we can solve the required
differential as,**

**The solution of the above linear first-order
ordinary** **differential equation**

**Now by substituting the value on the above solution, we
get,**

**On solving the above integration we get,**

Substitute back the value of ** v**, we
get,

**Step 2.) By using the initial value**

we get,

Solution becomes,

**At t = 1, we get**

For the initial value problem, Use Euler’s method with a step
size of h=0.25 to find approximate solution at x = 1

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...

1. Consider the initial value problem dy/dx =3cos(x^2) with
y(0)=2.
(a) Use two steps of Euler’s method with h=0.5 to approximate
the value of y(0.5), y(1) to 4 decimal places.
b) Use four steps of Euler’s method with h=0.25, to
approximate the value of y(0.25),y(0.75),y(1), to 4 decimal places.
(c) What is the difference between the two results of Euler’s
method, to two decimal places?

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)?

Consider the initial value problem
dy/dx= 6xy2 y(0)=1
a) Solve the initial value problem explicitly
b) Use eulers method with change in x = 0.25 to estimate y(1)
for the initial value problem
c) Use your exact solution in (a) and your approximate answer in
(b) to compute the error in your approximation of y(1)

Approximate the value of e by looking at the initial value
problem y' = y with y(0) = 1 and approximating y(1) using Euler’s
method with a step size of 0:2.
Also, how do I know when you stop? Very confused on that.

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) =

For the system ?? ?? = −2? ?? ?? = 1 2 ? with ?(0) = 0, and ?(0)
= 1 .
a) Show that (?(?), ?(?)) = (−2 sin(?) , cos(?)) is the solution
to the initial value problem.
b) Use Euler’s Method with a step size of Δ? = 0.1 to find an
approximate solution.
Find the approximate values at ? = 5, 10, and 20. That is, if
?(?) represents the approximation to ?(?), and ?(?) represents...

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.

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)

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