Using Cramer's rule to solve [x + y = 8 lr - y = -2, we obtain and y = 3 2 4 5 1 1. 3 1 1 ||

Q: Part 1: 1 1. S *(In x)3 dx 2. Se3x cos x dx 3. S(1 + x²)e-*dx 4. So x²e*dx

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Q: _11. What is Sdx? A. +C B. +x + C х-1 C. c. ++ 2x + C x(2x2+3x+6) + C x3 D. _12. In solving S th dx,…

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Q: Question 4. Consider the equation Utt – Uxx = 0, x E R, t ER. (1) For which c E R does (1) admit…

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Q: 3. Solve the following initial condition problems a. f'(x) = sin(2x)+ cos(3x), f(0) = 5 b. g"(x) =…

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Q: Find the function f(x)  described by the given initial value problem f"(x)=2sinx, f'(pi)=-5,…

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Q: Solve the given initial-value problem. The DE is of the form cos(x + y), y(0) = dx =// = f(Ax+By+…

A: What is Initial Value Problem: An initial value problem is related to an ordinary differential…

Q: Write S, = E(-1)i-la; where i=1 in ai = (-1)–1 sin x dx. (i–1)ñ x Show that a; > 0 and (an) is…

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Q: 5. eу-2*(1 + x?)dx + сos y dy %3D 0, when x %3D 0, у %3D0 2e2* (сos y - sin y) + еУ(2x? + 2х + 3) %3…

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Q: By using first fundamental theorem of calculus. Solve the problem 1.) Suppose f(x, y) = S*3 If fx…

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Q: sin x – x dx = ? X 22n (a) E(-1)". 22n+2 (b) E(-1)". (2n).(2n)! (2n +2).(2n+2)! n=1 n=1 22n+1 22n-1…

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Q: a. Prove that the O.D.E: (2y+2x-5)dx+(6y+2x-1)dy=0 is exact and solve it.

A: First we will check if this equation is exact.

Q: Question 5 The general solution of the DE y" - 3y" +3y' - y=0 is Ye = c1 e* + c2 e* + c3 e* Ye = C1…

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Q: 31. (a) 32. (a) 33. (a) 34. (a) ₁12x-11dx • S²₁₂ √2+ |x|dx 2+ |x| L₁ let |e* - 1|dx *) ₁²³-1-15|dx +…

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Q: Solve the initial value problem. dy = f'(x) = x-x² +x-1, f(0) = 1 %3D %3D dx

A: initial value problem is Substitution of initial condition in a general anti derivative of a…

Q: 6. f(t) = , e* sin 2r dr C.. -s(s-1) (s-1)+4 -2 A. s(s+1)+4) В. s((s-1)+4) (s-1)2+4 D. B.

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Q: Find the function f(x) described by the given initial value problem. f"(x) = 3 sin x, f'æ) = 3, f(1)…

A: The objective is to find f(x), find the area of the region encloses by the parabola and the x-axis.

Q: Solve this problem in digital format (Not by hand) step by step Solve the initial value problem by…

A: By the method of Separable of Variable

Q: Solve the second derivative forf (x,y,z) = sin* (x² +y²) a²f ax? x²+y² sin ت )م(. (x² +y²' sin sin…

A: The given function is: f(x,y,z)=esinx2+y2z Differentiating it w.r.t. x: ∂f∂x=esinx2+y2z∂∂xsinx2+y2z…

Q: -) Prove that the given function is The general solution of the given equation a) k'y"+Ky'+y =o y=G…

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Q: 25- х2 In dx , if we let u=/25- x2 , then udu= – xdx. X A) True B) False dt can be solved by 3…

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Q: If z + In(e3* - e + e²) = 0 dz find at the point (0, 0) -3 -2 3. 2.

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Q: Question (1)  Prove , ez1/ez2  = ez1-z2

A: The given equation is: ez1ez2=ez1-z2

Q: f (x) = xe-* – sin?x Newton Raphson (1st Method) @ x1 = 0.5 Newton Raphson (2nd Method)@ x1 = 0.5…

A: Newton Raphson Method:- Hence by Newton Raphson Method Root of the equation is 0.61447 (correct…

Q: Find the function f(x) described by the given initial value problem. f"(x) = 2 sin x, f'(T) = -4,…

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Q: 2) Suppose A is a constant such that Ae2a is a solution to the DE y' + 4y + 3y = e2a. Find the…

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Q: Question3: A. Find the general solution of the following first-order ODE xy' = y + 2x³ sin² (²)

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Q: Q5/ Solve the gamma function 2-3* dx 360 A) 2/n B) 2 V3 In 2

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Q: Show that r n2n-1 r=0

A: Want to  prove:    ∑r=0n rnr=n2n-1

Q: Question 37 1 Solve the integral: S- dx √√√√x+x A m/1+ + X +C ℗ 6n/5+x²³| +C B Ⓒ1+x²+c с X C =…

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Q: from Example 10.18. Using Runge-Kutta method of fourth order, solve for y at x = 1.2, 1.4 dy 2xy +…

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Q: Question 1 Prove that 4 = ucy)e is a Solution for d24 - y°4 = 0 dy?

A: Given differential equation is d2ψdy2-y2ψ=0         (1) We have to check ψ=u(y)e-y22 is the solution…

Q: Question 7 3 Calculate the integral: / 2 19 - x² dx 27n а) 2 b) О Зл 27n 4 9n d) 2 e) O 9n

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Q: What is a suitable integrating factor that can be used to solve the equation: X +2у — 4ху - е *cos…

A: topic - integrating factor for differential equation

Q: e* sin(e^)ax 8+cos²(e*) (x+2)dx x2-3x+5 .2 dx

A: Given that

Q: '0 (Fhop (4) (d) det(4) is non-real. (e) det (A)

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Q: e) S„ =E(-1)". sin (5 +n5) Vn+1 e) S. = n=0

A: Note:-  Sum of convergent and divergent series is divergent. cos2x=1-2sin2x⇒sin2x==121-cos2xput…

Q: Find the function f(x) described by the given initial value problem. f”(x)=8sinx, f’(pi)=-5,…

A: Given f''(x)=8sinxUpon integrating we get:∫f''(x)=∫8sinxWe know that ∫sin x=-cos x⇒f'(x)=-8cos x +…

Q: Question 3 хаx S- can be solved by either algebraic subsitution or integrataion by parts. x- 1 T…

A: The given integration,         ∫x dxx-1

Q: In 2 e*2arctan ex* J (1 + e2x) sec(2arctan e*)'

A: We need to evaluate ∫ln 2 ex 2arctan ex(1+e2x)sec(2arctan ex)dx                         (1) Here we…

Q: Suppose f(x) is positive, continuous, and f'(x) < 0 for all æ in [ – 8, 4], and| f(x)dx = 76 and L…

A: Concept: Mean value theorem: The mean value theorem states that if a function f is continuous over…

Q: d) 3/2 d.x (x4 +1)2 16 Cos(2.r) T e) dx %3D 1+ x2 e2 8. 8. sin x d.x (x2 + 4)(x - 1) Cos 1 - |

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Q: Chapter 1.1 Solve the ODE by integration or by remembering a differentiation formula. y' +e=/½ = 0…

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Q: 4s+5 b) Find L[eª“ f' (=cos 2) dt] and L-1 l(s-1)²(s+2)]

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Q: Question 5 4x2 Calculate the integral: dx 5+ x² V5 + x² - 10 ln V5 2х + C (5 + x²) 32 V5 + x2 V5 2х…

A: Indefinite Integration

Q: b- Compute the roof of the function f(x) = cos(x) +2 sin(x) + x². Please use xo = 0 and x1 = -0.1 as…

A: 4) Instead of x1,x2,x3 I choose variables x,y,z a)

Q: A linear, time-varying mathematical model is A. 4 (1– sin 2t) + 2x² = 0 dx B. 4 + 2x2 = sin 2t C. 4+…

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Q: If* f(x) = xe, then Newton's formula has: (A) Xn+1 (B) Xn+1) (C) Xn + 1 (D) Xn + 1 Xen Vehicle-1…

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Q: e -3 E y, Ax2 e-3* cos(4x) + Bx2 e4* sin(-3x) F yp = [Ax³ + Bx2 + Cx]e-3x

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Q: Theorem 8.3.3 If X ~ t(k), then E(X2r + 1) = 0 and µz, = k"- (2r – 1)(2r – 3) ... .1_ (k– 2)(k – 4).…

A: Answer: For the given theorem, The proof is:

Q: Question 10 x²+x+1 Ja 2x²+2+x³+x So - dx = [Aƒ (x) – B ln(g(x)] + D fb dx, and if A = ,B<0, then…

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