Question

Consider the initial value problem

dy/dx= 6xy^{2} 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)

Answer #1

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?

dy/dx = x^4/y^2
initial condition y(1)= 1
a) use eulers method to approximate the solution at x=1.6 and
a step size od delta x = 0.2
b) solve the differential equation exactly using seperation
variabled and the intial condtion y(1)=1.
c) what is the exact value of y(1.6) for your solution from
part b.

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

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.

solve by the integrating facote method the following initial
value problem
dy/dx=y+x, y(0)=0

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

dy/dx = x^4/y^2
a) use eulers method to approximate the solution at x =1.6
starting at the initial condition of y(1)=1 and a step size of
delta x=0.2
b) solve this differential equation exactly using separation
if variables and the inital condition y(1)=1
c) what is the exact vwlue of y(1.6) for the solution found in
part b

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)
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt + (8y + 6t
− 1) dy...

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