Consider the DE 6 Σ(+1 i=0 d'y (i+1)x¹ = e* - sin 2x. dxi a) Its order is b) Its degree is c) How many initial conditions are needed to find a particular solution for the DE?

Q: If +a= a(2 cos 0 cos 20) and y=a(2 sin0 sin 20),show that (z² + y² + 2az) = 4a² (z² + y²). %3D

A: Given that, x+a=a2cosθ-cos2θ⋯⋯1y=a2sinθ-sin2θ⋯⋯2 Squaring both sides of (1) and (2) and adding,…

Q: 4) g(n) = -n² + 3 = -11 A) g '(n) = Vn 1 2 + B) g '(n) = V C) g '(n) = Vn + 3 D) g "In) = 2n*

A: Solution ) g(n) = -n3 + 3 y = g(n) Swap y = n  n = - y3 + 3 n - 3 = -y3 y3 =  - n + 3 y= (-n + 3…

Q: (a) Write the equation in slope-intercept form, and determine the slope and y-intercept. (b) Graph…

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Q: aw аnd ar aw Q1) Find if as w = x2 + 3y + z, X =r - s, y= cos (r+ s), z=e'.

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Q: dx b) V2+ V3x² +1

A: Since you have asked multiple questions in single request so we  will be answering only first…

Q: Q1: Find 1) e (3 + e6t)2 2) l {e¬3t (t + 1)²} 3) е sin 3x dx dt 4) l (3t5) 5) e (est + cos 3t)

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Q: еху (ay cosica) since) -y + ay²e²) dx = ) dx = (x - 5.n²cx) - 4xye Ja

A: Exact differential equation Mdx+Ndy=0 [ where My=Nx ] has the solution. ∫M (taking y as a…

Q: Determine the order, degree, unknown function, and the independent variable of the DE 2x + 3x – 5x =…

A: in the first problem  order = 3 degree =1 unknown function =x(t) unknown variable = x independent…

Q: a) Use the d'Alembert solution to solve UTM UTM &UTM -00 <I < 00, t20, UTM 1Pu 9 dr u(r,0) = cos r,…

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Q: d y then is dx? x=1-8t Suppose y = 5t* +2

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Q: a) Express z = e(4+i(37/4)) in Cartesian form. b) Express z = cos(3i)+In(-3– 3i) in Cartesian form.

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Q: If y=t3, then y(6)+y(5)+y(4)+(1/3)y'=y? What is the power of y? a) 0 b) 2/3 c) 1/3 d) 3 e) 2

A: Differentiation of y with respect to t can be given as:

Q: Which one is not equal to o V(x + 3)³ dx. lim, 100 E i=n-1 (음 + 3)3. O lima-00 1ri=n-1 Li-0 2i+1 +…

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Q: udv – v(1 + v cos 2u)du = 0

A: We will find out the required solution.

Q: dx 2 sin X dy ☺ Cym) de + $cx-) dy =o $ (x-1) dy= o

A: (2) sin(x)dx+y2(x-1)dy=0⇒ sin(x)dx=-y2(x-1)dy⇒sin(x)(x-1)dx=-y2dyIntegrate both sides we…

Q: =dx V2+ V3x² +1

A: Since you have posted multiple questions, we will answer the first question. If you want the…

Q: Prove that M₁ = et/2 sin(B₁) is a martingale, by using Itô's formula.

A: Step 1: we can express  stochastic process M(t) as M(t)=∫A(t)dB(t)where B(t) is a martingale, then…

Q: 2 a) Solve x y'= (2+ yx)·

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Q: i) S'(2x – 1)* dx (In x)2 j) fth dx 7 k) S (1-- -)dx (x2+1) 1) S(1+ tan² x) dx

A: I am attaching image so that you understand each and every step.

Q: Q1: Find 1) l (3 + e6t)2 2) e {e-3t(t + 1)²}

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Q: = (e+ tan())“ then what is the value of y' (0)? a) 0 b) -1 c) 1 d) -2 e) 2 191 a) a b) c) C d) d.

A: To find the derivative of y

Q: A. J- sec (sec x) dx v xVx -1 elin dx c. Sa dx x +4 x + 5x – 6 2s+2 ds: D. JE+1)(5 – 1) B.

A: Since you have posted multiple subparts we solve first three subparts for you To get the solution of…

Q: (7) Show that the given functionis an integrationy factory Solves a) Sinydx+ Cosydy =0(e),…

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Q: 4. Find_r(t), if_r'(t)=4sin(2t)i +8t³j and r(0) = 8i-3j. A) r(t)=-2 cos (2t)i+2t¹ j B) C) r(t)=-2…

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Q: 1 b)y" + y = ; 1+sinx

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Q: Consider the DE y/" + (x sin æ)y' = y. a) Its i) order is ii) degree is b) Generally, when we have a…

A: The function f(x) is a rule that associates or fixes a number to the particular point x. If y=f(x),…