a) Without doing any calculations, state whether the vector field: er sin2(xy) F = -V 1 + x*y4 is conservative, and briefly justify your answer. b) Find the value of the constant k for which [k sin (xy) + xy cos (xy)] dx + xr cos (xy)dy is an exact differential.

Q: 5) Consider the conservative force field F(x, y) = (6x²y? + cos(x), 4x y). a) Give a general…

A: For a vector field F=P,Q that is open and simply connected in a region with continuous first order…

Q: 3. L{y"' + y' + 2y = sin 3t} ; when y(0) = 0, y'(0) = 0, y"(0) = 0

A:

Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F>0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: 1. Find the work done by the force field, a. F(x, y) = x²i – xyj in moving a particle along the…

A:

Q: 10) Find the potential function for the vector field F=-i-4j - k. z² Then compute the path integral…

A:

Q: 92 that the elect ric potential prove -5x V = e Cos 13y sinh 12 Z achieves the Laplace equation اثبت…

A:

Q: Consider the vector-field F = (2 — ysinr − 1) + (cos x − y ). – - (a) Show that this vector-field is…

A:

Q: Use the differential operators y'"-5y'+6y=e* %3D

A: Given that a second order non homogeneous differential equation and find its solution. Here i…

Q: 5. Find the potential function for given vector field: F = (e* In(y))i + + sin(2))j + ycos(z)k

A:

Q: Find the outward flux of the given field across the given cardioid. 5x i+ (e*+5 tan 1ly)j 1 + y2 F =…

A: The following formula from Green's theorem is used to find the outward flux, outwards flux=∮F·n…

Q: The magnetic field along a solenoid is conservative and given by 2xy 7- (2 x ₁-2x, (x²+1)-2). ¡−2x,…

A: Given a vector field as F=2xyx2+1-2x,lnx2+1-2 We need to find the scalar potential from which this…

Q: a) z = 4x²y-sen(3x - 4y)

A: # As per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: atch the differential equatio

A: (i) Given,          y'=2-y

Q: 2. Calculate the directional derivative of the scalar field U(x, y,z) at the point M,, along the…

A: Given that : U=xz3y-x3y=xz3y-x32y12 The gradient will be calculated as :…

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

A:

Q: | 23. Find the outward flux of the field F = ( 3xy i + (e* + tan¯ly)j х 1 + y?, across the cardioid…

A:

Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

A:

Q: 23. Find the outward flux of the field i + (e + tan'y)j 1 + y², F 3xy %3D across the cardioid r =…

A: Given: The following formula is used to find the outward flux, Comparing with the general form,…

Q: Show that the vector field F = x²i + (2yz)j+ y² k is conservative and then find the corresponding…

A: A vector field F is conservative if curl(F)=0 or ∇×F=0 The potential function is given by f such…

Q: Is the vector field F(x, y) = (sin(x)ely=3 + 2r cos(r>)y°, -2e)y-3 + 6 sin(x³)»*) conservative?…

A: Consider the given vector field.

Q: 1,1,0 ex 0,0,0 Show that the vector field defined under the integrals e koct and that the scaler…

A:

Q: A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The…

A:

Q: Calculate the work done by the vector field F(x, y)= ,2xy) on a particle that moves once around the…

A:

Q: Is the vector field F(x, y) = (6ry cos(3x²y + e*) +e* cos(3x?y + e*), 3x² cos(3x²y + e*))…

A:

Q: du d'u cos(-t) du 2- + u(x) = Cos I-2 du dr? sin(a-t)dt, rp u(0) = 0, u' (0) = 0. %3D

A:

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

A:

Q: (iv) Find the work done by the force field F(x, y) = x²i+ yeªj on a particle that moves along the…

A:

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…

A:

Q: Find the curl of the vector field F = (3x sin(y), 7y cos(x)).

A: The solution is:

Q: Determine if F = (12r – 2x cos y, 2y + x² sin y) is a conservative vector field. If it is, find a…

A: Given that F=(12x3-2xcosy)i+(2y+x2siny)j. We need to determine whether if F is conservative field or…

Q: For what values of a and d does the vector field F = ⟨ax, dy⟩ have zero flux across the unit circle…

A: A vector field, have zero flux. Across the unit-circle centered at the origin and oriented…

Q: To model the spread of a disease in a population of size N, the differential equation model == (kb -…

A: aTo model the spread of a disease in a population of size N,thedifferential equation…

Q: The vector field A is given by A = 2r cos(r* + 3y")i + 6y cos(2² + 3y*)j. a) Show that A is…

A:

Q: Find the counterclockwise circulation and the outward flux of the field F = (-sin y)i + (x cos y)j…

A:

Q: Express the field D = (x2 + y)-(xax + yay) in cylindrical components

A:

Q: 10. Show that the function f(x, y, z) = e³x+4y cos 5z satisfies the Laplace equation.

A:

Q: 43 Find the work done by the force field F(x, y) = (x² ln(y + 5), -) on a particle 3(y + 5) that…

A: Given force field F(x,y)=x2ln(y+5),x33(y+5) and circle x2+y2=4. The objective is to find the work…

Q: Calculate the line integral in scalar field ∫C (x2y2-√x)dy is the arc of the curve y=√x of (1,1) to…

A:

Q: (a) Express the vector field D = (x² +y<)-1 (xax – yay) in cylindrical components and cylindrical…

A: Given vector field: D =x2+y2-1  xax-yay a) To express the vector field  in cylindrical components…

Q: Find a linear differential operator that annihilates the given function. (Use D for the differential…

A: We have to find a linear differential operator that annihilates the given function.

Q: The integral of the vector field F(x, y) = (cos |x|, sin y|) A = (-2,0) and final point B (0, 2),…

A: Correct answer is (B) sin(2) - cos(2) + 1 Explanation:

Q: du du du + u(x) = cos r- 2- cos(-t) d'u dt-2 / sin(a-t) dt, 2 dt dr? u(0) = 0, u'(0) = 0.

A:

Q: Consider the conservative vector field given by: F (x, y) = (x - ycos (x), y - sin (x)) A potential…

A:

Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

A: Consider the given field: F→(x,y)=-yi^+x2j^ and

Q: Compute the divergence and curl of the vector field cos (22 ) sin² (z² yz) 10 yz4 – xyz2 at the…

A: We have to find divergence and curl of vector field:

Q: Use Laplace transform to solve the ordinary differential equation with discontinuous force. 10 sint…

A: Here, we will be solving the ordinary differential equation with discontinuous force by the method…

Q: Express the vector field D = ( x2 + y2 )-1 ( xax + yay ) in cylindrical components and cylindrical…

A:

Q: 4. Given the vector field Č = p² sin o a, + (z² cos – 1)as + (p² – z)a, Determine the x-component of…

A: Given the vector field

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.