Question

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 the approximation to ?(?), from Euler’s method then give (round to 10 decimal places):

(?(5), ?(5)) =

(?(10), ?(10)) =

(?(20), ?(20)) =

Please show steps and explain clearly. i need to progress my understanding

Answer #1

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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...

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)?

Solve the following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

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

(1) Consider the IVP y' 0 = y, y(0) = −1.
(a) Estimate the solution using Euler’s method with n = 2
divisions over the interval [0, 1].
(b) Carefully compare your approximation graphically to the
actual solution values.

Apply Euler’s method in the interval [0,1] with the step size of
0.2 to solve numerically the initial value problem: , . Report the
approximation for , and all the intermediate steps. y' = x + y/2
y(0) = −8

(b) For a given function u(t), explain how the derivative of
u(t) with respect to t can be approximated on a uniform grid with
grid spacing ∆t, using the one-sided forward difference
approximation
du/dt ≈ ui+1 − ui/∆t , where ui = u(ti). You should include a
suitable diagram explaining your answer.
(c) Using the one-sided forward difference approximation from
part (b) and Euler’s method, calculate the approximate solution to
the initial value problem
du/dt + t cos(u) = 0,...

Let (1，1) be the initial approximation of a solution of
(x + y) sin(xy) = 1
(x - y) cos(x^2y) = 1:
Use the Newton's method to find, and report, the next two
approximations of the
solution.

Problem 6. Use Euler’s Method to approximate
the particular solution of this initial value problem (IVP):
dydx=√y+x satisfying the initial condition y(0)=1 on the
interval [0,0.4] with h = 0.1.
Round ?? to 4 decimal
places.

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