a) Show that u(x,y) = x3 – 3xy2 + 3x2 – 3y2 is harmonic and find its harmonic conjugate

Q: (2) Let F(x, y) = (cos(y)+2x, -x sin y+1). Show that F is path independent by finding %3D a…

A: For the given vector function to be independent, it must be derived from a potential scalar function…

Q: Sketch the strophoid shown below. r= sec(0)- 2 cos(8), -<0 < x/2 1.0 1.0 0.5 0.5 -10 -0.5 0.5 1.0…

A: A strophoid is a curve generated from a given curve C and points A (the fixed point) and O (the…

Q: az az Use the chain rule to find and as at z = In(4x + 3y), x = s sin(t), y = t cos(s) az as az %3D…

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Q: determine where the given complex mapping is conformal. 3. f(z) = z + e + 1

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Q: Draw the bifurcation diagram for i = (x – 3)(c+ x)

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Q: t2 and y2 3 are two distinct solutions of the IVP t²y" – 4ty + 6y = 0, The functions yi %3D y(0) =…

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Q: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

Q: Evaluate f(x + 2y)dx + (y – 2x)dy around the ellipse C defined by x = 4 cos 0, y = 3 sin 0,0 < 0 <…

A: Given the Integral : ∮c(x+2y)dx-(y-2x)dy ⇒Let I=∫C(x+2y,y-2x).(dx,dy) , x=4cosθ and y=3sinθ…

Q: Let r(1) = (sin(t), cos(1), 2 sin(t) + 2 cos(21)). Find the projection of r(t) onto the xz-plane for…

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Q: 4. Find the harmonic conjugate of u where u(x, y) = x2 – y? – 2xy – 2x + 3y.

A: We will answer the first question since exact one was not specified, Please resubmit the question…

Q: An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y).…

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Q: 3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: See the attachment

Q: Find the solution u(t, x) to the system Jau +30₂u+u²=0 for t≥0, 20, u(t = 0, x) = cos(x). Start by…

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Q: 2- - Calculate the part a) using the given countour and assuming that R → o. For part b), use…

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Q: Write an equation for the conic in the xy-plane for - ) 1 at e- 30°. 15 3x° + 42/3xy- 39y² – 360 - 0…

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Q: (1) Verify that the real-valued function u(x, y) = xª – 6æ?y? + y* + 4x – 6y + 2 - defined on the…

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Q: Draw the bifurcation diagram for à = (x + 1)(x² – 2.x +r).

A: We are given : Now, the bifurcation diagram for this is :

Q: 2. Show that the Euler-Cauchy Equation Iy" + axy + by = 0, 1+0 can be transformed into the constant…

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Q: Show that u(x, y) is harmonic in some domain. u(x, y) = 2x – x³ + 3xy?

A: since Uxx+Uyy=0, then U is harmonic

Q: Si se utiliza la transformada de Laplace resolver la ecuación diferencial X"+ 5X' + 6X = 3t – 3e-3t…

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Q: determine where the given complex mapping is conformal. 5. f(z) = (z – 1)2

A: The complex function given is as follows:…

Q: Let (z-a)" dz where a € C and the parameterization of the curve is a = {|z| = r}. Apply the…

A: Hint: The statement of fundamental theorem  of complex analysis is: If a continuous function f has…

Q: 4. Classify and reduce the following equations to a canonical form: (a) yu -x*u, = 0, x>0, y>0. (b)…

A: To classify and reduced in canonical form of the given equation is..

Q: Show that the Euler-Cauchy Equation ry" + ary + by = 0, r+0 can be transformed into the constant…

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Q: и (х, у) 3D х3 +х — 3ху? What is the imaginary part of the analytic complex function if(z) = u(x, y)…

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Q: (b) Use Leibniz's product rule to derive the quotient rule y du- v dy d YY

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Q: Verify that u(x, y) = e®²-y´ cos(2xy) is harmonic in R?. Find the harmonic conjugates v of u, and…

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Q: az and as az Use the chain rule to find at z = In(4x + 3y), x = s sin(t), y = t cos(s) az as az at

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Q: determine where the given complex mapping is conformal. 4. f(z) = z + Ln z + 1

A: Solution..

Q: The Real Part of the complex function f (z) = u + iv is, %3D u(x, y) = 2x – 3x²y + y³ + 5 - (i)…

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: Show that U (x, y) = e"cos v,V(x, y) = e"sin v are conjugate harmonic of each other if f = u + iv is…

A: Solution given below

Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y)…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Let r (t) = (sin (f),cos (1), 9 sin (1) + 4 cos (2r)). Find the projection of r (1) onto the…

A: Given that, r(t)=sint,cost,9sint+4cos2t

Q: Find the harmonic conjugate of u where u(x,y)=x^2 - y^2 -2xy-2x+3y

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Q: (a) Consider the equation e'ux+2ayuy +2e*uy-sin(y) u =0, where a is a real constant. Determine the…

A: We will use basic knowledge of PDEs to answer this question.

Q: 2. 22 – x2 1 By sketching the traces on the ry-, rz- and yz-plane a.

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Q: Draw the bifurcation diagram for i = (x – 3)(c + x)

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Q: Express x = In(3t), y = 5 – t in the form y = f(x) by eliminating the parameter. %3D y =

A: Given that x=ln(3t) and y= 5-t we need to write y=f(x) by eliminating the parameter  Now , x= ln(3t)…

Q: Show that u (x, y) is harmonic in some domain and |find a harmonic conjugate v(x, y) when y (a) u(x,…

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Q: 2. Let u(x, y) = 2+ 3.x – y + x² – y? – 4.xy. Show that u(x, y) is harmonic and find the harmonic…

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Q: find Fourier sertes [o, 2w] an くxく下 O f cx)= (2 2下-X πく*く 2x fcx) = こ 2.

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: Show that the “conic” x2 + 3y2 − 6x + 12y + 23 = 0 has no points.

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Q: 2. Consider the PDE u -t ux+ tu =0_for - ∞ <x< ∞ and 0 <t< ∞ with u (0, x) = sine x. Determine u (t,…

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Q: Let z = g(x, y) = x² + xy³ -y - 6x - 2y +4, where x = r cos (8) and y = r sin(8). дz Use the chain…

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