Question

B1. Consider the initial value problem given by y 0 (t) = sin2 (y) ln(t + 1), with the initial condition y(0) = 1. Perform three iterations of Midpoint Method, using a step size of h = 1 3 , to obtain an estimate of the solution at t = 1.

Answer #1

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) =

Given :
dy/dt = -100000y + 99999 e^(-t)
a. If y(0)=0 use the explicit Euler method to obtain a solution
using a step size of 0.1. Carry out 2 iterations. What do you
notice? b. Estimate the minimum step size required to maintain
stability using the explicit Euler method.
b. Repeat the problem using the implicit Euler method to obtain
a solution from t=0 to 2 using a step size of 0.1.

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...

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: y' - (7/2)y = 7t + 2e^t
Initial condition: y(0) = y0
a) Find the value of y0 that separates
solutions that grow positively as t → ∞ from those that
grow negatively. (A computer algebra system is recommended. Round
your answer to three decimal places.)
b) How does the solution that corresponds to this critical value
of y0 behave as t → ∞?
Will the corresponding solution increase without bound, decrease
without bound, converge...

for the given initial value problem: (2-t)y' + 2y
=(2-t)3(ln(t)) ; y(1) = -2
solve the initial value problem

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...

Consider the initial value problem given below.
y' = (x+y+1)2 , y(0)= -1
The solution to this initial value problem crosses the x-axis at
a point in the interval [0, 1.4]. By experimenting with the
improved Euler's method subroutine, determine this point to two
decimal points.

Find the solution of the given initial value problem: y " + y =
f(t); y(0) = 6, y' (0) = 3 where f(t) = 1, 0 ≤ t < π 2 0, π 2 ≤
t < ∞

Consider the initial value problem
y' +
5
4
y = 1 −
t
5
, y(0) =
y0.
Find the value of
y0
for which the solution touches, but does not cross, the
t-axis. (A computer algebra system is recommended. Round
your answer to three decimal places.)
y0 =

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