d^2y/dx^2 − dy/dx − 3/4 y = 0,
y(0) = 1, dy/dx(0) = 0,
Convert the initial value problem into a set of two coupled
first-order initial value problems
and find the exact solution to the differential equatiion
Consider the first-order separable differential
equation
dy/dx = y(y − 1)^2
where the domain of y ranges over [0, ∞).
(a) Using the partial fraction decomposition
1/(y(y − 1)^2) = 1/y −1/(y − 1) +1/((y − 1)^2)
find the general solution as an implicit function of y
(do not
attempt to solve for y itself as a function of x).
(b) Draw a phase diagram for (1). Assuming the initial value y(0)
=y0, find the interval of values for y0...
1.) (10pts) Consider the following differential equation:
(x^2)(dy/dx)=2x(sqrt(y))+(x^3)(sqrt(y))
a)Determine whether the equation is separable (S), linear (L),
autonomous (A), or non-linear (N). (An equation could be more than
one of these types.)
b)Identify the region of the plane where the Chapter 1 Existence
and Uniqueness Theorem guarantees a unique solution exists at an
initial condition (x0, y0).
2.(12pts) Consider the IVP: y'+y=y/t , y(2) = 0
For each of the functions y1(t)and y2(t)
below, decide if it is a solution...
1). Find the dervatives dy/dx and d^2/dx^2 , and evaluate them
at t = 2.
x = t^2 , y= t ln t
2) Find the arc length of the curve on the given interval.
x = ln t , y = t + 1 , 1 < or equal to t < or equal to
2
3) Find the area of region bounded by the polar curve on the
given interval.
r = tan theta , pi/6 < or...
Consider the following first-order ODE dy/dx=x^2/y from x = 0 to
x = 2.4 with y(0) = 2. (a) solving with Euler’s explicit method
using h = 0.6 (b) solving with midpoint method using h = 0.6 (c)
solving with classical fourth-order Runge-Kutta method using h =
0.6. Plot the x-y curve according to your solution for both (a) and
(b).
Consider the following second-order ODE: (d^2 y)/(dx^2 )+2
dy/dx+2y=0 from x = 0 to x = 1.6 with y(0) = -1 and dy/dx(0) = 0.2.
Solve with Euler’s explicit method using h = 0.4. Plot the x-y
curve according to your solution.
Consider the series X∞ k=3 √ k/ (k − 1)^3/2 . (a) Determine
whether or not the series converges or diverges. Show all your
work! (b) Essay part. Which tests can be applied to determine the
convergence or divergence of the above series. For each test
explain in your own words why and how it can be applied, or why it
cannot be applied. (i) (2 points) Divergence Test (ii) Limit
Comparison test to X∞ k=2 1/k . (iii) Direct...