Define stpes to obtain the general solution of a 2nd order homogeneous ODE with real coincident roots?

Q: Find the formula for the function represented by the integral dt 6t+3 %3D

A: use the substitution method  for finding the integral

Q: Prove that three dimentional volume element "dx d ment is invariant under Lorentz transformations.

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Q: Compute the Cauchy principal value of the following integral using residue theory details): 2+1 dz…

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Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

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Q: What is the resulting integrable form of evx+I x+1

A: We have to find value of given Integral.

Q: Evaluate the integrals using fundamental theorem. | (2 sint – 24) dt

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Q: Evaluate the integral. 5 cosh(t) dt

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Q: Evaluate the definite integral exactly, using the Fundamental Theorem. 6. 1 -dt 1 dt II

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Q: Use an appropriate local linear approximation to estimate the value of the given quantity. In (0.98)…

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Q: Find the definite integral using the Fundamental Theorem. 2 1 dt 4t NOTE: Enter the exact answer. ~2…

A: Definite integration problem

Q: Compute the definite integral as the limit of Riemann sums. .2 3) (x2 + x + 1) dx

A: Definite integral

Q: Use a second Taylor polynomial at x = 0 to estimate the areaunder the curve y = ln(1 + x^2) from x =…

A: Given that, y=ln1+x2 To estimate the area under the curve from x=0 to x=12, use the second Taylor…

Q: Evaluate the improper intergal 1 dxusing the residue theorem. (x2 + 9)?

A:

Q: Identify the type of generalized integral +00 fx'e* dx- and in continuity (if converging) to be…

A: According to question,Given Integral,Let I= ∫0+∞ x2ex3 dx

Q: Find the area bounded by the curve y = x² – 6x + 5 and the x-axis from -2 to 2 using %3D Riemann…

A: area bounded by curve using Riemann sum

Q: / (In z)*dz. Evaluate the integral

A: Here we will use integration by parts repeatedly.

Q: Evaluate the integral 9t tan (1) dt

A:

Q: Consider the definite integral. (8x- – 1) e*x²-x dx Let u = 4x2 - x. Use the substitution method to…

A: Answer and explanation is given below

Q: Perform a test to determine if it converge or diverge -1 x 2sin 1 dx 1– x -1

A: The function is given by: ∫-112sin-1x1-xdx If the function fx=2sin-1x1-x is positive, continuous and…

Q: Find the local linear approximation of f(x) = In(5 + x) at xo = -4. %3D In(5 + x)

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Q: At the point of discontinuity on (0, co), the Fourier Cosine integral corverges to: f(x+) – f(x¯)]…

A: We have to solve given problem:

Q: dx%3D 9+x²

A: Solution is given below:

Q: (2- secx) secx sec(3A – x) eIn (xle) What should be the value of A for to be integrable using the…

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Q: Evaluate the Integral / tanh (2x)đx. A @ --n|cosh(2x)| + C B -In|cosh(2x)| + C © 21n|cosh(2x)|+ C D…

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Q: sec =zydy

A: Given

Q: Evaluate the iterated integral: dydx - 1/4 Jx2

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Q: Consider the following ODE with boundary conditions y" + λy = 0, y(0) = 0, y' KN ㅠ = = 0.

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Q: Given

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Q: Evaluate the integrals ∫et cos (3et - 2) dt

A: We have to evaluate the integral: ∫et cos3et-2dt In this case we should use substitution method…

Q: Apply Residue theorem to find the value of the integral siniz² (z+1)²(z-2)' C:lz + 11 = 3.

A: We have to apply the Residue theorem to find the value of the following integral : ∫C sin πz2z+12z-2…

Q: Show each derivative step in generating the fourth order Taylor polynomial for ln(x) centered at…

A: Topic = Taylor series

Q: Use the local linear approximation of (1 – x)2º at xo O to estimate (0.99)2º. %3D

A: The linear approximation of f(x) at x0 is given by L(x) = f(x0) + f'(x0) (x-x0)

Q: Evaluate the integral. inegral  ax / (x2 - bx ) dx

A: Consider the given integration as I=∫axx2-bxdx⇒I=∫ax-bdx

Q: Find an approximation to the integral integral 0 to 4 (x 2 -3x) dxusing a Riemann sum with right…

A: Given that: The integral is ∫04 x2 - 3xdx and n = 8.

Q: Find the Selution of (1-t +3t)4 (t) di - , (2) In which IL) denoter Bessel function

A: Leibniz integral rule : ddx∫bxaxFtdt=∫bxax∂∂xFtdt+Faxddxax-Fbxddxbx   Linear Volterra Equation of…

Q: Let f(x) = (cosh x)² be defined on (0, 0), find %3D d 1/2) . dy

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Q: Consider the definite integral. (10x - )x – 1) ex²-x dx Let u = 5x2 – x. Use the substitution method…

A: As per our company guidelines we are supposed to do only first question. Kindly repost the other in…

Q: Show that the following line integral is path independent an (3, 2) S cosh(+ X (0,0) (8xdx+6ydy)

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Q: Compute the integral using contour integration: 27 cos no (cos 0+a)20 for a > 1.

A: Note:- ∫C fz dz =2πi ×sum of residues cosnθ= Re ei nθ  cosθ=ei θ+e-i θ2      Given that  ∫02π…

Q: Use Cauchy's Residue Theorem to compute the improper integral (x + 3) sin(x) dx (x2 + 1)(x² + 4)2 00…

A: Given that ∫-∞∞x+3 sinxx2+1x2+42dx Using Cauchy residue theorem, we can written it as ∫-∞∞z+3…

Q: 1. What should be A for the path independent of (2,4) xy + A 2Ay – x + xp- y? fip- (1,2) Moreover…

A: According to the given information, it is required to find the value of  so that the below integral…

Q: /2 etan y sec y х cos" ydxdy

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Q: Evaluate the iterated double integral. I 3 3y be dady A

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Q: D Find a singular saluhion of DE: %3D

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Q: Consider the definite integral. (18x – 1) e9x²-x dx Let u = 9x? - x. Use the substitution method to…

A: We have to find the integrand after substitution.

Q: Use an appropriate local linear approximation to estimate the value of the given quantity. In (0.99)…

A:

Q: sec² (1+ /7) dr. 6) Use the substitution u = 1+Va to evaluate the indefinite integral|

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.