Prove using Mathematical Induction that: The sum of the squares of the first n positive integers is equal to n(n+1}{2n+1)/6: 1²+2²+3²+...+n² = n(n+1)(2n+1)

Q: 2 1) Solve : (D² − 6D+9]y = x²e³x cos2x -) Solve : (x² D² − xD + 4)y = cos log x + x sinlog x :)…

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

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Q: Find the solution of the given initial value problem. ty' + 3y = t² − t +5, y(1) = 6, t > 0 y =

A: We solve the given IVP  ty'+3y=t2-t+5....(1) y(1)=6.....(2)

Q: e terminal side on the unit -:-)

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Q: approximate value for y(3) for the following I.V.P. 2x + 1 y' = = y (1) = 0 5y4 + 1'

A: find value of y(3) using euler method

Q: Question 1: Solve the following differential equations using Laplace transform method. dy a) +7y=5…

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Q: Q4) Use separation of variables method to solve: d²u ди ax² + =u where u(0, y) = 0 and ду du(x,0) by…

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Q: The mean weight of five complete computer stations is 167.2 pounds. The weights of four of the…

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Q: (a) (i) ◄ Q8 (b) char(Z₂ x Z7 X Z₁4) = 14 (c) Z₂ x Z7 x Z12 = Z14 × Z12

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Q: span0001 a. 2b+c O & [[ 2b + c b ]:b,CER} 0 Ob.{[-2b + c b ]:b.CER} 0 Oc.{[-2b-c b]:b,CER} 0 d.[[ 2b…

A: 1) The span of the two matrices 2100 and 1001 is the linear combination of these matrices. span…

Q: Problem: The concentration of salt x in a home-made soap maker is given as a function of time by: dx…

A: For a differential equation of the form: y'=f(x, y), Euler's method uses the formula:…

Q: 5 By separation of variables, Solve the following PDES- ²4 = 4 24 dx² dt² 0 o u(e,t) = u(s, t) =6…

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Q: 1- In the library, the prices are as follows: The pen is $5, the notebook is $7, the book is $30..…

A: “As per the policy we are allowed to answer only 1 question at a time, I am providing the same.…

Q: 1. · (y')² + (x − 2x¯¾)y' + x^² - x = 0

A: This is the problem of the differential equations.

Q: b) Let T:R³ R¹ be a linear transformation, and let B = {e1, ez, e3} be the standard basis for R³.…

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Q: Write the equation of the line that represents the linear approximation to the following function at…

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Q: A = SooC 0 3 -8 0 0 0 9 T 1

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Q: emon-crusted salmon (L),

A: Given that,  L = Lemon Crusted Salmon C = Crab-stuffed Chicken G = Garlic prime rib W = Wasabi rolls

Q: 2. Consider the following four functions from R into R, determine if onto or 1-1 function. Support…

A: To find out which of the following function is one-one,  onto .

Q: PART#C Use the divergence theorem to calculate the flux of the vector field P(x, y, a)-ar+…

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Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

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Q: Let be the vector field defined on the plane by F(x, y) = (x¹+3y, 6x - siny).

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Q: Which among the sets is open? [0,1) O {0.1/n...1/3,1/2,1} R\[a,b] None of the above.

A: To identify which of the given set if open set.

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A: Find the flux of the vector field Fx, y, z=-3xy i-1yz j-5xz k through the sphere S of radius 4…

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Q: The wave equation = may be studied by separation of variables: u(x, t) = X(x)T(t). If(x) = -k² X(x),…

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Q: 39. Z z²+9

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Q: Q2) Find the complementary (homogeneous) solution of y"" - y = f(t). Determine the particular…

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Q: (c) Given the set T := {(x, y) = R²: |xy| ≤ 1}. Is T a compact set? Show your working. If you say it…

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Q: 1 0<x< 1/ 9=2-1<x<** (b) Find Fourier cosine transform of f(x) = {

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Q: (e) For any nonzero real number z, prove Σε 1- 1-z""

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Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: 7. Yoshi is experimenting with an image-editing program on his computer. He rotates his picture…

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Q: Which among the following diagrams correspond to the metric below on the real plane R^2?…

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Q: (d) A commutative ring R is a prime ideal of itself. (e) If p and q are primes, then there is a…

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Q: Solve y′′(t)+2y′(t)+2y(t)=H(t); y(0)=1 ;y′(0)=0 where H(t)=2t+3 , 0≤t<1 H(t)=3t+1, t≥1

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Q: Suppose that A and B are finite sets. Prove that AxB is finite.

A: Result 1 :  A be an non-empty finite set if and only if cardinality of A is a finite natural number…

Q: 1. f sin(u-v)y(v)d(v) + y(u) = 10u 2. x"(t)-5x' (t) + 10x (t) = e-3t, x(0) = 0, x'(0) = 0

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Q: 2. 3. NOR A NAND A OR A AND A NOT A NOT A NOT B NOT B NOR A NOT C NOT C MAND B NAND A

A: Theory: OR gate: A OR B ⇒ A+B AND gate A AND B ⇒A.B NOT gate NOT A ⇒A' NOR gate: A NOR B ⇒ (A+B)'…

Q: yees). You are free

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Q: dy dx =y-x, y(0) = 2

A: Introduction: The formula of the fourth-order Runge-Kutta method is given by,…