Use D'Alembert's formula to solve the following (IVP): Utt in R x (0, 0) | u(x, 0) u(x,0) = sin (x) in R = tan-1(x),

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: Find the extrema of f(x) = 2 sin x − cos 2x on the interval [0, 2 ].

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Q: 1₁ -1.2 - St 0 √x sin ex dx

A: Given :   I1=∫01.2xsinex3dx

Q: Solve the D. E. to the I. V. P. sinx(e + 1)x'- cosx 1, y() = 0 O (1 + e') (1 + cos x) = 2 O (1 + e)…

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Q: If u(t) = (sin(4t), cos(2t), t) and v(t) = (t, cos(2t), sin(4t)), use Formula 4 of this theorem to…

A: Find u(t) . v(t) first then differentiate with respect to “t”.

Q: Use the modified Euler Method to solve the ODE = subject to y 1 when x 0 %3D dx 1+x on the interval…

A: To find y(0.3) by using the modified Euler's method.

Q: Eliminate the arbitrary constant/s of this equation to obtain the DE. Express the DE in general…

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Q: /hich of the following does not require the use of the dy nain rule to find ? dx A y = cos1 (10x –…

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Q: Use Secant method to to approximate the solution of f(r) = z+ cos z =0 accurate to within 10-3,…

A: The secant method is used to calculate the root of the function. The formula for the secant method…

Q: (t – sin t)v1 – cos t dt

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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: 2.xy x² – y? Show that - = 0 if z= Tan

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Q: a) For 0 ≤ x ≤ 1, show that < x². √2 + x b) Use the estimate in a) and integrate it to prove that…

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Q: a. The velocity of a car in meters per seconds travelled from its origin is given by v(t) = In (t2 +…

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Q: For the following integration / sin"(24x) cos(24x)dx what is the suitable substitution? a.…

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Q: The approximation of I = , tanx dx using composite Simpson's rule with n=5 is: 0.6158 0.6165 None of…

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Q: Prove: ex. a. ſ csch x = In b. S sech x = 2 Arctan(e*) + c

A: a).  ∫cschx dx=∫2ex-e-x dx=∫2ex-1ex dx=∫2exex2-1 dx Put, ex=t⇒ex=dtdx⇒ex dx=dt Therefore, ∫cschx…

Q: Gren thint sin (A+i6) = xtiy, 28 Prove that, 9父+ =1 snh'B 1) x essec?A -y' sec'A =1

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Q: The substitution u = tan |converts the integral 2 π/3 1 dx to: n/2 2 +sin x - coS X

A: Given,∫π22π312+sinx-cosx

Q: (Find the integral curve of the following equation passing through the given point). y = 4 cos(x),…

A: Given dy/dx and point we have to find equation from this

Q: Find the temperature u(x, t) in a rod of length L, if the initial temperature is f (x) = 100 sin 6nx…

A: Given function is, fx=100sin6πx at x=0 and x=L Now, we know that, ∂2t∂x2+qgk=1α·∂t∂z....(1)

Q: Using the substitution t = tan(), the integral 2 dx sin(x) – 78 cos(x) can be written as dt 39 – t-…

A: To make substitution and re write the given Integral.

Q: The integral O [se Ss S S sec²0 de 1+ tane sece tane de sece - 1 O x dx x²+√√x²-1-1 sine de 1+ cose…

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Q: Suppose A(t) = [ e^t 0 0 0 cos t sin t 0 −sin t cos t]…

A: Given: At=et000cos tsin t0-sin tcos t To Find: To show that At-1 exists and then to find it.

Q: The approximation of I = , tanx dx using composite Simpson's rule with n-5 is: None of the choices O…

A: f(x) = tanx Each interval h=(b-a)/n =1-0/5=1/5 =0.2 xO=0, x1 =1/5, x2=2/5, x3=3/5,x4=4/5, x5=1…

Q: 5. If z = f(x, y), where x = r cos 0 and r sin 0, find z and and hence show that: (). (-(( 2 2 dz dz…

A: To find the partial derivatives of the following function in two variables by chain rule.

Q: DIfferentiate the following fuh ction with respect to K In X %3D Sin a

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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: The integral dx (x²+4)² sec ³0 de sin ²0 de ° csc ³0 de ° Scos²0 do can be transformed to which of…

A: Here we have to transformed the given integral.

Q: Let I = 3x*cos (x) dx. The approximation of / using the two- formula is:

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Q: The integral S (A S- (в x dx x² + √√√x²-1- sino do 1 + cos sec²0 do 1+tano " cose de 1-sino sece…

A: Use trigonometric substitution

Q: Use the Newton-Raphson Method to approximate the root of f(x) = 2 cos(x) – 2x + 1. (b)

A: Numarical methods: Newton Raphson method

Q: S sin x In(cos x) dx

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Q: 2. Find a closet rank-1 approximation in the f norm to A = cos 8 - sin e sin e cos 0

A: According to the given information it is required to find closet rank-1 approximation in the l2 norm…

Q: Find the value of the improper integral or state that it is divergent. (If it is divergent, enter…

A: We will calculate the definite integral using basic definite integral properties and if we get an…

Q: Evaluate the integral / cos (r) sin(x) dx 16 COS using the substitution u = cos(x). ,16 cos" (x)…

A: This question is based on indefinite integral.

Q: Let f(x, y) = 2e sin(4) of д х of dy Check Answer

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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: (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: With the substitution u = sin(x), we get | 11 sin2(x) cos(x) dx = 11 u2 du. u? du. 11 %3D which…

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Q: Write out Taylor's formula around 0 of degree three, f(x) = P3(x) + E3(x), for f(x) = = COs x.

A: Given that fx=cosx. To determine the Taylor's formula around 0 of degree three. That is, fx=P3x+E3x…

Q: 3. If w-f(x.y) and x-r.cose, y-r.sin6 Prove that - A + fy ar

A: Given that w=f(x,y) and x=rcosθ,y=rsinθ Prove that ∂w∂r2+1r2∂w∂θ2=fx2+fy2

Q: Obtain the approximate answer for 2 cos(x )dx using the trapezoidal method so 3) that the maximum…

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Q: Express T(x)sin x-root(3)cos x into the form of T(x)=Asin(x+theta) , where A>0

A: Sin(A+B)=sinAcosB + cosAsinB

Q: If u(x, y) = 2 tan-1 2 and v= In (x² + y?) , prove that: a2v = 0 дхду dx2

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Q: F (t)= 7tei+ sin²(2t)j – In (5t)k Vhat is F'(t)?

A: Here we apply detective.

Q: Use the second fundamental theorem of calculus along with the chain rule to find d sint dt dx 1 o…

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