Consider the curve given by the equation y^2 - 2x^2y = 3
a) Find dy/dx.
b) Write an equation for the line tangent to the curve at the
point (1, -1).
c) Find the coordinates of all points on the curve at which the
line tangent to the curve at that point is horizontal. d) Evaluate
d 2y /dx2 at the point (1, -1).
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...
Consider the differential equation dy/dx = y^2 + y - 2 (1)
Sketch its phase portrait and classify the critical points. (2)
Find the explicit solution of the DE.
1. (a) Sketch the slope field for the given differential
equation: dy/dx = 2?
(b) Find the particular solution of the differential equation
that satisfies the initial condition y(0) = 4
(c) What is the value of y when x = 1/2
2. (a) Find the general solution of the given differential
equation: dy/dx = ysinx = ????? 2
(b) Find the particular solution of the differential equation
that satisfies the initial condition ? = 2; ?ℎ?? ? =
π/2