1. f(x,y)=x²+y² is a homogenous furion of degree 2

Q: 14. Show that the function f: R²R given by if (x, y) 0, ²+y² f(x, y) 0 if (x, y) = (0,0), is…

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Q: i) Find an extremal y(x) of the functional I(y) = S,y' – y)²dx, y(0) = 0 y(1) = 2

A: Functional: Let S be a set of functions. Let f : S→ℝ be a real-valued function. Such functions are…

Q: Determine the extremal of the functional: J = | [x²(t) + u²(t)]dt With boundary conditions x(0) = 1,…

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Q: Show the function f(z) = z (z) is conformal at 0 and f'(0) = 0. Does this function violate our…

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Q: The extermals of the functional J[y] =√²² (y¹¹² y² + x²) dx sunder conditions: ex y lot=1,…

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Q: (3x2-2xy+3y2)dx = 4xydy Find the general solution by determining the type of differential equation.

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Q: Consider 6x (1+x²)² 2(x+y) (1+x²)² + 2y ý = Using the function V(x, y) = 12 + y² show that the…

A: What is Lyapunov Stability Theorem: Lyapunov stability theorem says that if a function is positive…

Q: Let Ly)= x²y"- x(x+2)4'+ (X+2)y as Consider the homogeneous diffeuntial equation, Lly)=0 (*) is show…

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Q: 8. (a) T F The xy-trace (on the xy-plane) of the surface x² + y² − z² = 0 is a point. (b) T F If…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: 4. Consider the function g(r, y) = re 2r2-2y. (a) Determine and classify the critical points of g(r,…

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Q: 7. ( ) The surface area of the surface S with equation z = g(x, y), where (x, y) lies in D and f has…

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Q: 7. So -y dx + x dy x² + y² G₁₁: x² + y² = 1 C₁ traversed in a clockwise direction. (b) Green's…

A: The given integral is ∫C1-ydx+xdyx2+y2 ;  C1: x2+y2=1 Parameterize the curve as  x=rcos θy=rsin θ…

Q: dx dy dx +γ = 2(c-Ay) and Y. + ds ds ds where c and d are constants and rt 8(t) = √6° Show that…

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Q: 7. Show that the system L' = 3y + 2x – r(4x² + 2y²), / = -3r + 2y – y(4.x² + 2y?), has at least one…

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Q: 2. Consider the DE y(4) – 4t²y" + 4y" = 0. (a) Is the DE linear? (b) Is the DE homogeneous? Why? (c)…

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Q: Consider the differential dz 2.ry da + (x² + 2y) dy. (11.39) %3D Evaluate the integral S2 dz along…

A: The integral, ∫(x1,y1)x2,y2dz=∫(x1,y1)x2,y22xydx+(x2+2y)dy∫(x1,y1)x2,y2dz=∫(x1,y1)x2,y2Mdx+NdyWhere,…

Q: (2) Calculate the flux of F(x, y, z) = (x², y², z²) outward across the boundary of the domain D =…

A: Here S is the boundary of a sphere which is closed so we can apply Gauss Divergence Theorem.

Q: Find the holomorphic function f (2) = u(x, y) + iv(x, y) (z = x + iy) with f(0) = 0, given that v(x,…

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Q: Consider the IVP: 4y"(t)-4y'(t) + y(t) = 0, y(0) = 0, y'(0) = 1 (a) The characteristic equation is:…

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Q: | Let ƒ(x, y) = 4x² + 7xy + 4y² + 7x − 7y + 4. Find the linearization L(x, y) of f(x, y) at the…

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Q: y" + 4y' + 3y = 0, y(0) = 2, y(0) = -1

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Q: Consider the non-homogeneous third order equation y''' − 4y'' + 5y' − 2y = 30 e^t . A. Find a…

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Q: Let f(x, у) %3D 7у — 8х + 2ху+ x3 — у?. a) Find the gradient of f at the point (1, –1). b) Find the…

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Q: 2. a) Find the general integral of the equation zwa+ywy = ryw in the domain {(x,y): >0, y>0}. b)…

A: 2)(a)The given equation is  and the domain is..Find the general integral as follows.It can be solved…