Question

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?

Answer #1

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)

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 following initial-value problem.
(x + y)2 dx + (2xy + x2 − 2) dy =
0, y(1) = 1
Let af/ax = (x + y)2 = x2 + 2xy +
y2.
Integrate each term of this partial derivative with respect to
x, letting h(y)
be an unknown function in y.
f(x, y) = + h(y)
Find the derivative of h(y).
h′(y) =
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt...

1)Consider the following initial-value problem.
(x + y)2 dx + (2xy + x2 − 2) dy =
0, y(1) = 1. Let af/ax = (x + y)2 =
x2 + 2xy + y2. Integrate each term of this
partial derivative with respect to x, letting
h(y) be an unknown function in y.
f(x, y) = + h(y)
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt + (8y + 6t
− 1) dy...

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.

Consider the initial value problem
dy dx
=
1−2x 2y
, y(0) = − √2
(a) (6 points) Find the explicit solution to the initial value
problem.
(b) (3 points) Determine the interval in which the solution is
deﬁned.

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)?

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

Solve the initial-value problem.
(x2 + 1)
dy
dx
+ 3x(y − 1) = 0,
y(0) = 4

Solve the 1st order initial value problem:
1+(x/y+cosy)dy/dx=0, y(pi/2)=0

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