II) Consider the following three equations ry-2w=0 y-2w² <-2 <<=0 5 <=0 1. Determine the total differential of the system. 2. Represent the total differential of the system in matrix form JV=Udz, where J is the Jacobian matrix, V = (dr dy dw)' and U a vector. 3. Are the conditions of the implicit function theorem satisfied at the point (x, y, w; 2) = (1,4,1,2)? Justify your answer. w² +2²-

Q: (c) Show that the set of solutions of the inhomogeneous system is given by S = {(s, t, 3-s+t): s₁ t…

A: d) Given  that The,   also  can be written as  Clearly, S describes  a  plane  which is  not  passes…

Q: A system of the type i = f(x), f(x) = -gradV(x), is called a gradient system. Investigate the…

A: Given that, A system of the type x˙=fx, fx=-grad Vx is called a gradient system.  Investigate the…

Q: æ'(t) = x · (1 – a²), y'(t) = -y*

A: Now our first goal is to calculate the equilibrium point, so that we equate the given system of…

Q: 1. Write down the ODEs defining the streamlines of the time-independent (steady- state) flow u(x, t)…

A:

Q: dz dt dw dt = = 3ezw + 8 sin(z) = cos(82)

A: We have given system of differential equationsLet, Jacobian matrix is given by

Q: For vectors (x, y, z) € R³ consider the inhomogeneous linear system: x=y+z=3, and its 'associated'…

A:

Q: dx dt dy = −y − x(x² + y²)¹/2, = x − y(x² + y²)¹/2 dt (a) Show that the system is almost linear in…

A:

Q: Let g: R- R$ AND f: R3→R² be a c² function. g(5)=[3].f(1,3,2)=(3,0), g'(5)=(-2,2,2), and 2 -2 -1…

A: Given the functions g:ℝ→ℝ3 with g5=132, g'5=-222, and f:ℝ3→ℝ2 with f132=30, Df132=-2-1-225-2. We…

Q: 2. The behavior of the system is a saddle. x' = -x +y, y = -x-y

A: The given system of differential equations is:…

Q: Consider the system of equations a' = µx + y – x(x² + y³), y' = -x + µy – y(x² + y³),

A:

Q: Take the Van Der Pol equation x" — 6(1-x²)x' + x = 0. Write down the corresponding first order…

A:

Q: 36 3 For the function of (x,, ^₂^3) = X₁ X² & ²₂², determme the gradient vector of first-order…

A:

Q: Find a first order system corresponding to the scalar equation and find its vector form and general…

A:

Q: Consider the initial value problem of a system of linear differential equations x' = Ax+eAt "A], =…

A: The Given Initial value problem is: x'=Ax+eAt01 , x(0)=10, Where A is 2×2 Constant matrix. Let…

Q: 1. Determine the Jacobian matrix (matrix of partial derivatives) of the function: f(x, y, z) = (x -…

A: Step 1: Step 2: Step 3: Step 4:

Q: 9) Compute the matrix of partial derivatives of the following functions: @Jf: R² → IR², fox,g) =…

A: According to bartleby guidelines we do only first three subparts and rest part can be reposted…

Q: Exercise 5. (Fishery Model) The equation N = rN(1 –) – H represent a simple population model of a…

A:

Q: Find the fundamental matrix for the system: x'= 9x + 12y y'= -6x9y x(0) = -2, y(0) = -6 Fundamental…

A:

Q: 10.J Compute the matrix of partial derivatives of: f(x,y) = (x+y, x-y, xy) d.) f(x, y, z) =…

A:

Q: DTake the system a' = 8x – xy, y' = 6y + x2. %3D How many critical points are there? help (numbers)…

A: Given,x'=8x-xyy'=6y+x2For critical points put x'=0 and y'=08x-xy=0x8-y=0x=0 and y=86y+x2=0x2=-6yx=0…

Q: b) Plot the vector fields for different values of h (show all qualitative behaviors)

A: The given equation is, dxdτ=x1-x-h So, the vector field for different values of h are as follows.

Q: y "- 3y + 2y =0 0 1 O b. ***-(2,₂3). x-(2-1) --(-) 0 1 X' 3 2 21 +-(-3) X'= 0 a. O c. O d. x x

A:

Q: x' = x(3 − x+y) y = y(2+x-y), where x(0) ≥ 0, y(0) ≥ 0. Describe the global dynamics of the system

A: The given system of differential equations is: x'=x(3-x+y)  y'=y(2+14x−y),  Where x(0)≥0,y(0)≥0

Q: Let   dx/dt =   x2y3           dy/dt  = -x - y - (xy4)/2 be a dynamical system.   Is  L(x,y) =…

A: Given function is, dxdt=x2y3dydt=-x-y-xy42......(1) Lx,y=x22+x2y44.....(2) For the proper Lyapunov…