Suppose that r(t) = cos³ t i+ sin³ tj+et k, show that d dr (r.- dt dt d?r X at2 dr d'r. r.at at3 dt2) = r.(

Q: ду Let y = In Vx² + 2z3 + cos(b³), find

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Q: Q2/ Prove that 1 dx = v2 Jo 1+sin(x)

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Q: If u = x² cosy+ ye", where x = s+2t and y = st, prove that ди ди 2- (s – 21) (x siny- xye – e"). -…

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Q: that Find flz)= r°cos20 + ir?sin po Such %3D is analytic-

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Q: | cos zdr + sen ydy + x?dz

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Q: Show that u(x, t) = 4 tan-1 I-ut ev where v is constant, is solution of Uxz – Utt = sin(u)

A: To check the given solution satisfy the given partial differential equation, substitute the solution…

Q: dz dt Consider x = cos t, y = sint and z = x2 +y². Then ulan ||

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Q: Q7 Show that CX,y) =e is Condinuous at co,0)

A: The given function is:  fx,y=e-x2+y2 The given point is 0,0

Q: 2. If z y where z= s cos t and y=s sin t, find az/ôs and az/at.

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Q: b. Show that 0s cos ay dA s where R= (0. G ) sin Ax

A: This is a problem of double integration and inequality.

Q: In (T/e) ( (2- sec?x)eme sec³x sec (зА - х) -dx to be integrable using the formula и" du? ? What…

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Q: y = In(sec t), x = log(cosec t) In(sect), x %3D %3D

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Q: J. In (T/2) cos ey dx dy In (/6)

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Q: Assume that y is a continuously differentiable furction in t Evaluate a) tP (+),A (+)) d tant sect…

A: Given: y is a continuously differentiable function in t. (a) To find: ∫xexπy(t)y'(t)dt

Q: The integral (cos(3x)) sin(3x)dx under the change of variable u = cos3r becomes u du. Select one: a.…

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Q: Evaluate. d [arctan a – In ¤] dæ 1 1+#? 1 sec2 x sec² ¤ 1-x²

A: As we know that: The sum rule: f±g'=f'±g' The chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1…

Q: 00 cos wx Q1/ Use Fourier Integral to show that 0. dw w2+1

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Q: dy iven that x = e" cos 3t, y=e" sin3t, show that tan 3t + dx %3D 4

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Q: d - [sin(t) dt = 0, Where k is cons tan t dx Select one: True False x

A: Here, the integral has the constant limits 0 and k. So, ∫0ksin(t)dt s in terms of k that is…

Q: Find dy/dt, x=4 sin t; y=12 cos t* O 24 cos t O -24 sin t O 12cos t O -12sin t

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Q: A force of F(x)=x2−cos(3x)+2, x is in meters, acts on an object. What is the work required to move…

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Q: Suppose that w = a7 · exp(2y) · cos(4z) with z- in (+ 5), x = sin (t+ y = In (t +2), z = t.

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Q: The function z = r' y is defined on the astroid r = cos 0, y = sin 0 where 0 < 0 < 2ñ. Use the dz…

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Q: Find the linear approximation to g(z) = cos(z + 1) at z = 0. Use the linear approximation to…

A: The formula to find the linear approximation Lz of a function fz at the point z=z0 Lz=fz0+f'z0z-z0

Q: Evaluate the integral (10 + sin t)* cos t dt 14 using the substitution u = 10 + sin t. 14 10 + sin…

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Q: Evaluate the integral / (4 + + sin t)° cos t dt using the substitution u = 4 + sin t. (4+ sin t)°…

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Q: Let z = x?y + 3xy+ where x = sin 2t and y dz find = cos t. Use the chain rule to when t = 0. dt

A: See solution below. Using definition of chain rule.

Q: Suppose that w = x² + y? + z², x = cos 0, y = sin 0, z = tan 6. Use the chain rule to find dw when 0…

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Q: Let y=9 In(cos(a). in(co(e*). dy = AeBt sinC (e6t) Then dt cosº (e6t) where A = B = C = D =

A: Given expression y=9lncose6t Differentiate the above expression w.r.t "t" dydt=9ddtlncose6t Recall…

Q: (b) If u =x cosy+ ye», where x=s+2 and y= st, prove that 2- ds du - (s - 21) (* siny-xye -e").

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Q: Let C be defined by 7(t) = |cos (t), sin (2t), t²| for 0 < t < n 2 sin(Vz) 9x*dx + 2zdy + + 2y ) dz…

A: Using the result that  ∫cF·dr=∫cPdx+Qdy+Rdz=∫P(r→(t))dxdt+Q(r→(t))dydt+R(r→(t))dzdt dt Here,…

Q: Use the convolution theorem to show that cos T sin (t – T) dr 1 sin t. %3D 2

A: Convolution theorem use integration of product of 2 functions and transforms them to pointwise…

Q: Utt || for 0 < x < T u(x, 0) = x? u:(x, 0) = }cos(8x) Uz(0, t) = u„(r, t) = 0

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Q: Let T = x²y - xys + 6x, where x = r cos 0 and y = r sin 6. Use the chain rule to find dT/00.

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Q: If u =x² cos y +ye», where x =s+2t and y= st, prove that ди 2- Əs ди 3 (s— 21) (х' siny — хуе" —…

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Q: Let z = x*y + 3xy* where x = sin 2t and y dz find = cos t. Use the chain rule to %3D when t = 0. dt

A: z=x2y+3xy4where x = sin 2t and y = cos t We need to find dzdt when t =0 Let's use the chain rule :…

Q: H.W. using the relation r = VT show that/(x) = COS X %3D 2

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Q: Suppose w = In(8x² + 2y² + 3z²) , x = 9 cos(t) , y = 9 sin(t) , and z = 8/t. Compute dw/dt in two…

A: To find: The derivative of w and the value of dwdtt=8  Given: Suppose w=ln8x2+2y2+3z2, x=9cost,…

Q: (b) If u = x² cos y+ye", where x = s+2t and y = st, prove that ди ди 2 = (s – 2t) (x siny– xye –…

A: u=x2cosy+yexyx=s+2ty=st

Q: Q1: show that1) d/dz [cosz] = -sin z, %3D 2) 1+tan2 z = sec2 z

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Q: Calculate the Cauchy principal value. 8 COS X V.P. Leta -dx, (a > 0). (a²-x²)

A: We solve by using residues.

Q: Al/ compute 6(4+ ec©*) dx + (siny + 3x?)dy cosx Where c is closed bounded region lying in the first…

A: Given (A) The given integral is ∮4+ecosxdx+siny+3x2dy and the region lies in between x2+y2=4 and…

Q: a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian.

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Q: r(t) = cos(nt)i – In(t)j+ vt – 16k; to = 17 | -

A: Given :

Q: Prove that d/dz sin(z) = cos(z) using Cauchy-Riemann equations

A: Cauchy-Riemann Equation- If f(z)=u(x, y)+i v(x,  y) is analytic, then In particular, These…

Q: If w = Vr + y² + zª where r= sin(0), y = cos(@) and z = Use the Chain Rule to find tan(@). dw when…

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Q: Let z=In(x²+y²) where x=ue"sin t and y = ue"cos t. If u=- 2 and 1=0 then is equal to %3D ....

A: Multivariable function: When a function f has more than one independent variable and only one…

Q: find ds/dt. s = sin t/1 - cos t

A: Given    S=sin t1-cos twe have to find dsdt

Q: The solution of the DE (1+ cot'x)y' = cscx coty is %3D None of these siny + In|cosx| = C cosx + In…

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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.