Derive the following formulae (i) (ii) (iii) Σ Π r=1 έρ r=1 = Σμ r=1 r= n(n+1) 2 n(n + 1) (2n + 1) 6 = n(n+1) 2 .

Q: Q3 Consider the following second-order ODE: (2 – x)xy" (x) + (x² – 2)y'(x) – 2(x – 1)y(x) = -x².…

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Q: H.W. find (ti+7j+ (t+ 1)k)dt

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Q: (a) Let Þ(y) = $(t;y) = y +ty. Show that t → Pt(y) is the solution of the ODE for some p: {X'(t) =…

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Q: Find the general solution of the following ODE. y(4) + 4y" = 0 %3| y = c1 + c2(0)+ c3 cos(O) + C4…

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Q: Find an explicit form of the general solution of the following ODE: (x – 1) y" + 2y' = 0.

A: Given

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Q: Find f(t). f(t) = = -πS/2 s² +4 r{se se + X ) u (t-1/2 ₁

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Q: on f(x)

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Q: solve Laplace L-1( 2s −1/s2(s + 1)​​​​​​​3 )

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Q: By making a suitable substitution, find the general solution y= y(r) of the ODE dy = (y – 4r – 1)².…

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Q: Determine, with justification, whether the series converges absolutely, converges conditionally, or…

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Q: +3Y=e

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Q: 1. y = ao Σm=o(−1)m(2m + 1)x2m + ai Σ=o(-1)"(m + 1)x2m+1

A: 1+x2y''+6xy'+6y=0

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Q: Which of the following is the general solution of: y(x+1) 2 dy dx x Select one: O a. 2lny = x - ln x…

A: Given problem :- 2dydx=yx+1x

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Q: Consider the following ODE: d²x dt² The solution of this ODE may be chaotic. + (1+ sin(t)) da + x² =…

A: Given a system of (ordinary) differential equations ∆, the system is chaotic if there exists initial…

Q: 0 < n 4 (-a)n-1 ζ n=1 Σ +∞o IP sof Γ

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Q: If f(x)=14-x^(2) and g(x)=\sqrt(x+31), compute g(f(3)). Simplify your answer as much as possible.

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Q: Evaluate: L || e" 3t cos 2τdτ 4 s-3 s [(s – 3)² + 4] s[(s – 3) + 4] | s-3 1 (s – 3)² + 4 (s – 3)' +…

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Q: 1. a. Let In = exp (nx) sin (exp (x)) dx. By making an appropriate substitution, or otherwise,…

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Q: Q1: (1) Show that the value of € (-1, π), for which: sin² 0 1+ sin² 0 sin²0 1 + cos²0 cos² 0 cos²0 4…

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Q: Find r(t) that satisfies the initial condition(s) r″(t) = −4 cos tj − 3 sin tk,    r′(0) = 3k,…

A: Given r″(t) = −4 cos tj − 3 sin tk,    r′(0) = 3k,    r(0) = 4j

Q: s² – b? - L{t cos bt} s2 + 62 (s² + b²)² ° - (**- -

A: We have to find the Laplace transform of the following - Ltcos(bt)

Q: [t5 ] = ? %3D 120 24 s6 O A D QI L[(t+ 1)?e] = ? 24 120 s5 Oc B s2 + 1 s2-1 L [sin? (t) + cos?(t)] =…

A: Given 1 Lt52 Lt+12et3 Lsin2t+cos2t

Q: 40. Regression analysis makes use of the relationship between two or more quantitative variables so…

A: ...

Q: 20 x(t)= e 10 nt- u-n)_¸ ¯-¡(10π-÷n) 2j x (1) = -√ [6 / (10 mt - ² m ) _ _ - - ( 10 mt ² x(t) == é…

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Q: Find the Annihilator of the given function(s) f(x) = cos(2x) + 4sin(3x) a. (D²+3)² b. C. d. D² -3D+4…

A: answer is b

Q: Consider the following ODE system: = x² + xy³ =zy subject to the initial conditions x(0) = 2 and…

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Q: Let x(t) =u(cos(2xt)) - (u(t) -u(t-2)) and h(t) =eu(t), and let y(t) - x(t) h(t). What is the value…

A: Given that, xt=ucos2πt.ut-ut-2ht=e-tutyt=xt*ht

Q: S e3(2e * – 1 ) dx (ear – 2) (e3* + 1) 4 (A) 2e3x – 1 B) 2e 6r – e 3x © (e3r – 2)+ D (e*+1)+ (E ,6x…

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Q: 4. (a) Find the general solution to the following ODE. dy r+y² + xy² +1 dr (b) Given that y(0) = 1,…

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