ㄑㄨ ind all solutions. Also, plot a direction field and some integral curves
#8
In Exercises 7-10 find all solutions. Also, plot a direction field and some integral curves on the indicated rectangular region. 7. C/ Gy' = x(1 + y2); {-1 5*31, -15 y S1) 8. CIG y'(1 + x2 + xy = 0; -2<x<2, -1 3y 1}
3.7: The following is a direction field showing solutions to a system of differential equations for four different initial conditions. Each of the four time series graphs below correspond to one of the four solutions shown in the phase plane. Match the solution curves with the time-series plots. (a). (b)
1. (20 points) Let
(a) Determine and plot the equilibrium points and nullclines of
the system.
(b) Show the direction of the vector field between the
nullclines
(c) Sketch some solution curves starting near, but not on, the
equilibrium point(s).
(d) Label each equilibrium point as stable or unstable depending on
the behavior of the
solutions nearby, and describe the long-term behavior of all of the
solutions.
"ind all solutions to the equation on the interval (0,2) - 2 cos sinx+1=0
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2-y) (1 -y). Find all equilibrium solutions, label all pertinent coordinates. Note: An Autonomous equation means that dy/dt does not depend on time t. Hint: Follow the method demonstrated in Example 1.3.6 (p.28). The hand-draw slop field is optional and not necessary. This method gives a qualitative analysis for the future of all possible solutions without solving the equation quantitatively
15 pts] Sketch some representative...
Show by example that a field F' of quotients of a proper subdomain D' of an integral domain D may also be a field of quotients of D.
(a) Draw a direction field for the given differential equation. (b) Based on an inspection of the direction field, describe how solutions behave for large t. All solutions seem to approach a line in the region where the negative and positive slopes meet each other. The solutions appear to be oscillatory. All solutions seem to eventually have positive slopes, and hence increase without bound. If y(0) > 0, solutions appear to eventually have positive slopes, and hence increase without bound....
Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y' = 7 + 7y y / / / / / / / / / /3 / // // IX х 1/-0.2 // 0.2 10.4 -0,4 -0.2 20,4 / / / / 0.2 0.4 / 1 1 +3 1 +3 y y 13 1 11 1 1 2 / / / / / / / / / / // х -0.4 -0.2 0.2 0.4...
sketch sine integral curves.
. T 101111111111 1111111111 11111111111111 1111111111111 111111111111 111111111111: 111111111111 111111111!* 111111111 *** 1111/r **** 111111!!++++ 1111111!***** 1111111 17111rN +++++ 1 A direction field for y' = 1 1 1 / 1 / 1 1 1 1 1 TTT TTTTTTTT 1 1 1 1 i i i / rrrrrrrr **rrrrrrrrrrrr/ -02 TTT
Question 2. Consider a surface S in the 0 plane with three smooth boundary curves C1, C2, and C3 as shown in the diagram. Each curve is parametrised so that it is traversed in the direction shown by the arrows For a smooth vector field A(x, y, z) you are given the following results: Ca Adr =-3 C2 0.5 2.0 1.5 -0.5 (a) What is the value of the surface integral ▽ × A. ds. if we assume by convention...