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.