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: 31²+1 dt = Compute:

A: Dear student as per our rule in one session we can only solve one question. Please post the other…

Q: 2. By hand, sketch the following against independent variable t: a) x₁(t) = Re(2e(-1+j2m)t) b) x₂(t)…

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Q: a) Find the points at which the function f: C → C defined as: f(x + yi) = x³ − y² + (x − y)i - Have…

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Q: 6. Let F = (4e" cos(4x), e sin(4x) + 2y) be a vector field and let C be a curve in the first…

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Q: Let f"(t) = /t – 2 cos t. Determine a general f(t). -

A: Given : f'''t = t - 2cost

Q: 1. Find the derivative of the following: 1² + 2r + 1 a) f(1) = r3 - 5x + 3 b) G(2) - VP+1 dt e) y…

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Q: If flix.y)=XSin²(y)+y°Cos(x) prove that fn=2[Costy).Sin(y) ySin(x)]

A: First of all, we partially differentiate function f wrt. x and then we partially differentiate the…

Q: calculate r'(t) and T(t), and evaluate T(1). r(t) =〈cos πt,sin πt,t〉

A: Given :                   r(t) = cos πt , sin πt , tDifferentiating with respect to t, we get…

Q: If r(t) = = cos(-3t)i + sin(−3t)j – 10tk compute r' (t)= i+ i+ (cos (-3t))/3 and fr(t) dt= (-sin…

A: r(t)=cos(-3t) i+sin(-3t) k-10tk

Q: 1. Compute d e (z²) (sin(t) + e*)v= dr

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Q: what is the acceleration at time t=o of a particle if its position is s(t)= cos (3t)-sin(4t

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Q: Example: Sketch the direction field for the equation y' = y – t over the square -2 < t, y < 2, then…

A: The direction field is a graph of small line segments passing through various points whose slope is…

Q: Xexs is The Laplace inverse of s2+4x? O a. sin(2x(t-1)) 5(r-x)dr 0.

A: Let f(t)=L-1xs2+(2x)2           =x L-11s2+(2x)2          =x.12 sin(2xt) ∵L-11s2+a2=1asin(at)…

Q: Let f (t) = te* +e sin 10t. Find L{f (t)}.

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Q: Calculate :| t' sin(t) dt [Hint: At some point in the solution, it might be helpful to write t as t2…

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Q: Consider this ordinary differential equation (ODE): dy da At the point (-0.5, 1.5), the direction…

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Q: A yoyo bounces up and down in the vertical direction with its acceleration after tt seconds given by…

A: d^2y/dt^2 = a(t)

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

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Q: Let y= Sin x + 2 (05 X ) Ivaluate y, y'sy" and y'" at 4

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Q: Suppose r' (t) = (cos(t), et, 2t) and r(0) = (1, 0, 3). Find r(t).

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Q: dr = wa x b dt r X

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Q: Check if y = sin(t)et is a solution to: d2y/dt2 + dy/dt = et[3cos(t) + sin(t)]

A: We need to check if y = sin(t)et is a solution to: d2y/dt2 + dy/dt = et[3cos(t) + sin(t)]. We have y…

Q: Show that whether if the co-vector w can be written as dø differential? w = y. (x +). dx + x. (+).dy…

A: Given:  w=y·x+y2· dx+x·x2+y·dy....(1)

Q: Show that y1(t) = cos(3t) and y2(t) = sin(3t) are linearly independent for all t

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Q: c) Let f(x, y,z) = cos (xyz)sin(x²y=

A: The given function is f(x,y,z)=cos2(xyz)sin(x2y2z2) ..............(1) we have to find;  fx=∂f∂x=?and…

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.