. (a) Write cos 3x + i sin 3x = e3ix = (cos x + i sin x)³ by Euler's formula, expand, and equate real and imag- inary parts to derive the identities cos x = } cos x + cos 3x, sin x = } sin x – sin 3.x. (b) Use the result of part (a) to find a general solution of y" + 4y = cos³ x.

Q: )3--find all solutions of the equation 2sin? c-cosx=1 in the interval [o, 27).

A: To solve the following trigonometric equation  2sin2x - cosx = 1

Q: 4. Prove that the equation x + sin(x) = 1 has at least one solution in the interval |3D Va + 3 (0,…

A:

Q: 7.Obtain the general solution using the method of undetermined coefficients. a. (D² + D)y = - cos x…

A: First we put auxiliary equation D2+D=0 then solve it.

Q: 1. Find the general solution of y'= cos²xcosy

A:

Q: 5. Find the general solutions of the equations i) Zx - 4z,, – 5z, +10z, = cos(x+3y)-sin(x+3y) %3D

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2) find the radius and coordinate center for the equation 2x42y+10y-12х-20-0 3) prove that sin 2x…

A:

Q: Given that Y1 = x-1/2 cos x and y2 = x-1/2 sin x from a fundamental set of solutions of (--)- x²y" +…

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Q: Find all solutions to cos(7x)−cos(x)=sin(4x)cos(7x)-cos(x)=sin(4x) on 0≤x<2π30≤x<2π3

A: We know, cos(a)-cos(b)= -2sin(a+b)/2*sin(a-b)/2....(i)

Q: 2. Determine one real root of 8 (sin x) e-² – 1 = 0 xo = 0.30 using Newton-Raphson | with %3D Method…

A: To solve the following problem, we use the Newton Rephson formula

Q: x³y"+xy'+4y = cos (2 In x)

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Q: 2. i) Find in radians the values of x in the interval 0°0 and 2 is and acute angle. Hence find the…

A: The problems with the trigonometric functions for value of x can be solved using the trigonometric…

Q: Find the general solution of y'=cos²xcosy

A:

Q: The minimum value of cos x + 3 sin x + 5 is: - V10e A 5 - V10- в 5+ V10 D -5 - V10

A:

Q: 1 0 x' = 6 1 -6 |x 4 6 1 X = c1 -4 e + c2e'| cos 61 + c3e' sin 61 6. sin 61 - cos 61 6. -4 e' + c2e…

A:

Q: Find all exact solutions on [0, 2n) of 8 sin?x + 6 sinx + 1 = 0. O Fine all exact solutions on…

A:

Q: 2- Find the largest possible interval in which the given DE has a unique solution tan(x)y" + xy'-y =…

A: Given equation is, tanxy''+xy'-y=x3 ,x0=-2,x0=5 We can write the given equation as,…

Q: π/2 1-z [²6² x cos ydzdydx = = [Show your solution.]

A:

Q: Solve the he o-3D+4)y(x) =x cos(2x) equation y,(x)=(6x+4)sin( *+4)sin(2)-cos(23) - y,(x) =(-6x-4)…

A:

Q: 2. Find a particular solution (without solving for the coefficients) of y(5) + y(3) = x³ – 3x +7+ x…

A:

Q: 9. Solve for value of x [0, 2 n ] 2 cos x + V3 = 0 а. b. 2 sin 2 x - 1 = 0

A:

Q: 8. If x * and (cos x)sin² x- 3 sin x +2 2 then find the general solutions of x.

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Q: 4- A possible form for a particular solution of y"+2y = cos(v2 t) is a) Acos(2 t) + B sin(/21) b)…

A: This question is related to differential equations, we will solve it using given information.

Q: iFind the general solution of the equation sin4x 19. Find the general solution of the equation cos2x…

A:

Q: The second-order difference equation Yk+2 - 3yk+1 + 2yk = 2 sin(3k) (4.164) has the homogeneous…

A:

Q: 5) By using the method of undetermined coefficients, if the term of (r(x) = 3x² cos 5x), the exact…

A:

Q: b) cos 5x = -0.7 %3D a) sin 2x = 0.3 Cosc 57) to.7 Sin (2x) - 03-0 Cos C5x)+0.7 x -a xis at Letu=…

A:

Q: Find the solution for cos(x)" - (1 + x²) – 2xy = with the conditions dx2 1+x2 y(1) = -1, y(2) = 2…

A: Given equation is  cos(x)d2ydx2-(1+x2)dydx-2xy=11+x2 with y(1)=-1 and y(2) = 2 h = 0.25 To solve  -…

Q: Q5. (a) (i) (b) (c) (ii) Express sin 2x Express cos 2x Show that in terms of in terms of sin x Solve…

A: Given problem:- (a) (i) Express  sin 2x in terms of sin x and cos X.(ii)  Express cos 2x in terms of…

Q: x = -30°, cos 2x = 1 - 2 sin2 x 1 + 2 sin 2 x 1 - 2 sin 2 x 1 - 2 sin x 1 + 2 sin x

A: This question is based on Proving a Trigonometric equation.

Q: List the roots of the auxiliary equation for a homogeneous linear equation with real constant…

A: according to our guidelines we can answer only first question and rest can be reposted.

Q: 2) Find all solution of the equations in the interval [0,2x]. a) 2sin 30+ 12 = 0 b) (cosx- 1)(sin x…

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Q: Find all exact solutions on [0, 2?). 2 cos2(t) − 3 cos(t) = −1

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Q: 3 f(x) = x° – cos(x) %3D х — - Solve this question by the Fixed-point method on the topic of…

A:

Q: 4. Find all solutions of sin x+ 2sin x cos x-0 if 0sxS2.

A: Note:  As per our company policy we need to answer only one question each time so, please repost the…

Q: (9) Verify that Yp = sin 2x is a solution of y"– y = -5 sin 2r and use it to find the general…

A:

Q: 4. Consider the problem of estimating a root of the equation cos(2x) - x = 0 via Newton's method.…

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Q: 2 Find the general Solution of yty = 2xcosx -CosX %3D

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Q: 10. Solve for t on the interval [0, 2π) (a) 2 cos² t-cost-1=0 (b) 2 sin²t-3 sint + 1 = 0 taol al (c)…

A: Given

Q: 1. Solve the hyperbolic function of the following. i. cosh (-2 + j3) in the form of a + jb ii. sinh…

A: some formula  coshx+jy=coshxcosy+jsinhxsinysinhx+jy=sinhxcosy+jcoshxsiny i.…

Q: 2. Locate the first positive root of f(x) = sinx + cos(1 +x³)- 1 where x is in radians. Use four…

A: (a) We need to find the root between 1 and 3

Q: 15 a Show that cos 30 = 4 cos 0 – 3 cos 0. b By substituting x = 2 cos 0, show that the equation x…

A: 15(a) The left side is cos 3θ=cos 2θ+θ=cos 2θ cos θ-sin (2θ) sin θ=cos2 θ -sin2 (θ) cos θ-2 sin (θ)…

Q: A mass weighing 32 pounds stretches a spring 2 feet. If the mass is initially released from a point…

A:

Q: The set of solutions of y' = -2e-* – 12 cos(4x) + 2 is given by: а) О 2е + 3 sin(4x) + 2x + C 3 b) О…

A: Note the integration of some functions: ∫eax dx=eaxa+C ∫cos ax dx=sin axa+C ∫xn dx=xn+1n+1 +C ;…

Q: (cos 6, sin 0) (0,1f G4) 2x (号) (올) 120 3 3n 60 135 45 5x 150 6 30 (1,0) (-1,0) 0,0 180,t 7x 210, 6.…

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Q: (9) Verify that yp = sin 2x is a solution of y" – y = -5 sin 2x and use it to find the general…

A:

Q: 7. Solve the equation for exact solutions over the interval[0,2x). 2cos' (2x)=1- cos (2x) %3D

A: Since you have posted multiple questions, we will solve only first question for you. To getremaining…

Q: Find the solution in the interval from 0 ≤ θ < 2π 2sin2 x - cosx -1 = 0

A: Given : 2sin2x - cosx - 1 = 0

Q: (-1)"2x2n+1 2n +1 1. n=0 2x2n+1 2. (-1)". (2n + 1)! n=0 (-1)*2²" x2n (2n)! 3. n=0 2" x" 4. n! n=0 A.…

A: According to question "we have to choose the Maclaurin serieswith correct function correctly"

Q: -1 Find y' where y = 22 sin-(cos (9 æ)) – tan (6 æ) Answer: -198 sin(9 x) 1 || V1-cos2 (9 æ) 1+36 x2…

A: To find y'

Q: 8. Find the positive root of the equation 3x - cos x - 1 using Regula Falsi method and correct upto…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…