1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find a potential function for it.

Q: -1 < x <0 The Fourier integral of f(x) = 1, 0<x<1 0, otherwise 00 (1 – cos(w)) sin(wx) dw lw 1. 00…

A: Given that fx=-1-1&lt;x&lt;010&lt;x&lt;10otherwise The Fourier integral of f(x) defined on -∞,∞ is…

Q: 3 e flow in the field F = 4i + 9j + 2k along the path C, where C is the curve r(t) = 7 cos 4ti + 7…

A: Solution

Q: 4.1 In aerodynamics and fluid mechanics, the functions u(x,y) and v(x,y) in f(z) = u(x,y) + iv(x,y),…

A: As per the policy, we are solving the first question. Please repost the question and specify which…

Q: By the odd extension method applied to the function f(x)=e-2=, where r > 0, the Fourier integral of…

A: Given function fx=e−2x where, x&gt;0 We have to find the Fourier integral by the odd extension…

Q: b) Use Laplace transform to solve the system of differential equations UTM dr UTM x = 4y, dt y+Bet…

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Q: 64. Use the finite Fourier cosine transform to solve the heat conduction problem t> 0, 0 0, where…

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Q: Which of the following is a potential function f for F(x, y, z) = (2x cos(yz). -x²z sin(yz) + 2yz?,…

A: The potential function of the function: fx, y, z is ∇f and it can be defined as: ∇f=∂f∂x, ∂f∂y,…

Q: Q5. Find the Laplace Transform of the funetions x(t) and yıt), which satisfy the coupled first…

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Q: Use the Laplace transform to solve the initial value problem. 3. x' + 2y' - y = 1; 2x' + y = 0; x(0)…

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Q: Use Laplace transform to solve the partial differential equation: d²u d²u du(x,0) = t> 0, 0<x<1,…

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Q: Find the potential function f for the field F = 9x i+ 6y j+ 6z k. A general expression for the…

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Q: Use Laplace transform to solve the partial differential equation: d²u du ди(x,0) t> 0, 0<x<1, u(x,0)…

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Q: Use Laplace transform to solve the partial differential equation: d²u du du(x,0) = , t> 0, 0<x<1,…

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Q: Find a potential function f for the field F. F = (5y + 2z)i + (5x + 3z)j + (2x + 3y)k A general…

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Q: 5) (a) Show that F = (8-14xy2 + 2ye²x) + (e²x - 14x²y) y is conservative. (b) Find a potential…

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Q: 4xy V = -2xz XZ

A: relationship between voltage (V)  and uniform electric field (E) can be given by  E→=-dVdr=−∇→V…

Q: Find a potential function f for the field F. F = (3y + 5z)i + (3x + 4z)j + (5x + 4y)k A general…

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Q: Find a potential function f for the field F. F= (4y + 2z)i + (4x + 5z)j + (2x + 5y)k A general…

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Q: Find a potential function f for the field F = ev *2=(4 i + 4x j + 8x k). A general expression for…

A: We can calculate the following question by usual method of calculating potential function

Q: 8. Prove that the differential operator M: C² [a,b] → C[a,b] defined by d²u du Mu= 2+3u dx is a…

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Q: 1. Let F(x, y, z) = (2re²*+3», 3e* +3v +2/E, ). 2.x cez²+3y a. Show that F is conservative using its…

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Q: Find a potential function f for the field F. F = (4y + 5z)i + (4x + 3z)j + (5x+3y)k A general…

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Q: the potential function f for the field F = 5x i + 9y j + 4z k. eral expression for the infinitely…

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Q: Use Laplace transform to solve the partial differential equation: ²ud²u du(x,0) t> 0, 0<x<1, u(x,0)…

A: Solution :-

Q: 13. (a) Sketch the slope field of dy dt = t/y in the region -2 <t< 2, -2 < y < 2. (b) Check that y…

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Q: Find a potential function f for the field F = 4y + 5z ( i + 4x j + 5x k). A general expression for…

A: Solve as general

Q: Suppose an undamped forced spring mass system is driven by the equation + 9x = f(t), dt where the…

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Q: Find a potential function f for the field F = 2y + 5z (4 i+ 8x j + 20x k). A general expression for…

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Q: compute CF d? for the veder field. KI+x3 xy> where Cis the line segment from CO.0) to Clo3)

A: The objective here is to calculate the integral: ∫CF→⋅dr→; Where F→=1+x2,xy2 And C is the line…

Q: Consider the following initial-value problem for u(x, t), du du = Au XER", dt Əx1 |u(x, 0) = f(x),…

A: As per the question we are given a partial spatio-temporal differential equation in Rn dimensions,…

Q: Which of the following graphs is the slope field for 15/1 1~/1ST ~\~ in 13t-ii-i ~~~ -1 1 2 3 4 5…

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Q: b) Use Laplace transform to solve the system of differential equations UTM&UTM UTM UTMUTM de dy - r…

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Q: 4. (a) Show that F (2xy + z³)i+x²j+3xz²k is conservative force field. Find the scalar potential of…

A: Please see the attachment

Q: 1. Sketch the direction field and solution curve for the given DE and condition. Sketch the…

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Q: T'y" - 5xy = sin 3ry is a second order nonhomogeneous differential equatic Select one: O True O…

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Q: 6. Suppose f(2) is analytic in |2| 1, and f(0) = 1, f(0) = 2, compute: %3D (1) h = fe(s+ 1)2d: (2)…

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Q: Let z= 28x + 32xy - 8y , where x(t,s) = cosh (4t) cos (5s) and y(t,s) = sinh (4t) sin (5s). ze at…

A: Here, z=28x2+32xy-8y2

Q: 13. Compute the Laplace Transform of f(t) = t°e" +e* sin3t. 4 1 A) (s+ 2)'s² +4s + 13 2 B) (s –…

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Q: 7. a) Show that F(x,y)= is a conservative field. b) Find a potential function ø(x, y) fa [show your…

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Q: Q2:- a. b. Let u=x²-2xy + 2y³ with x = s²ln t and y = 2st³. Find au/as and au/at. find the total…

A: We need to solve the two questions.

Q: Suppose L is a polynomial differential operator of order 2 and L(te^(3t)) = 5e^(3t), L(e^(3t)) = 0,…

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Q: Use Laplace transforms to solve the initial value problem x"'+x = cos3t; x(0)= 1, x'(0)=0

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Q: Suppose w = In(æ² + y² + z²) , Compute dw/du and dw/dv in two different ways: ue3" sin(9v) , y :…

A: GIvenw=ln(x2+y2+z2) where x=ue3vsin(9v) , y=ue3vcos9v and z=ue3vTo find∂w∂u and ∂w∂v

Q: Find a potential function f for the field F = e²y + 52 (2 i + 4x j + 10x k). A general expression…

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Q: 10. Show that the function f(x, y) = e=2y cos 2r satisfies the Laplace equation.

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Q: Find a potential function f for the field F = ey*2 (2 i + 6x j+ 10x k). +

A: First we will show that F is conservative Then finding the potential function by considering it as…

Q: How does the limiting behavior of solutions depend on the value of y(0)?

A: Given autonomus differential equation:Graph of function f is given.We need to show the the limiting…

Q: Given F(x,y) = &lt; 3x^2e^y, x^3e^y + 1/y &gt; is conservitive, Find the potential function phi(x,y)…

A: Given: F(x,y) = &lt; 3x^2e^y, x^3e^y + 1/y &gt; is conservative.

Q: Find a potential function f for the field F = e* * 3²( i+ 4x j+ 3x k). A general expression for the…

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Q: a) Without doing any calculations, state whether the vector field: er sin2(xy) F = -V 1 + x*y4 is…

A: (a) Since ∇×∇=0 So, curlF=0 Hence, F is conservative field.

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.