1. Consider the wave equation = Utt UrL = 0, u(t, x). (1) Ət² dx² , Given the (Lorentz) transform from t, x to tť', x': (t, x) → (ť', x') 1 ť = y(t – Bx), x' = y(x – ßt), (2) V1- 82" where B: real constant, 0 < 3 < 1, [better to use: u(t, x) = u(t(t', x'), x(t', x')) = u(t', x') instead of w(t', x') in the notation of the textbook]. a) Write the differential operators d/dt and d/əx interms of t' and x' using the Chain rule: 0/dt = (at' /Ət)(a/dt') + (dx' /Ət)(0/dx'), d/dx = (dt' /əx)(ð/dt') + (ða' / Əx)(0/dx'). b) Write this wave equation interms of the new variables t', x'. [You should conclude that this wave equation is covariant/invariant (not change its form) under this transform.]

Q: √²³₁(−12xy² + 8x³y + 3)dxdy

A: LetWe have to evaluate the above integral.

Q: ikt 三 ik ik Ikt. ik こ

A: So we have to solve the inverse Fourier transform to get the expression for U(x,t)

Q: Consider any(n) + an-1y(n-1) + + a1y' + aoy = f(x) where an, ...a1, ao E R, where an 7 0. ... (a)…

A: consider anyn+an-1yn-1+.....+a1y'+a0y=fx            ...i where an, ......., a1 , a0∈ℝ where an≠0 (a)…

Q: Solve the wave equation using Fourier Transform: o'u ou - 00 0 u(x, 0) = 0 u,(x, 0) = H(x) e %3!

A:

Q: Find the vector equation of the tangent line to the curve given by 7(t) = sinti + cos (2t)j + tan tk…

A:

Q: 2C

A:

Q: 3). y" + 2y' - 8y = 24, Given the IVP: y(0) = 0, y'(0) = -6. A) Use the Laplace transform to find…

A:

Q: (r,1) = (r sin (f), r– cos (1)), (r.1) = (5,-) s numbers in exact form. Use symbolic notation and…

A: In this question, the concept of the Jacobian of functions is applied. The Jacobian of Gr,t is…

Q: use Laplace trans forms g - 4 y + g'co)= o 3 y = 6t - 8 , ylo)=0 ,

A: Solve the following

Q: Determine the Laplace transforms of 3, 0 12

A:

Q: 2. The inverse Laplace Transform of the function F(s) = 4s+2 / s²+2s+5 is : (a) e* cos 2t + 2e* sin…

A: Given function is Fs=4s+2s2+2s+5

Q: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

Q: By using a suitable Fourier transform, the solution of 9u

A: In this question, we have to solve the wave equation by using Fourier transform. The wave equation…

Q: 7. If f is a scalar field and v = [v1, v2, v3] is a vector field, show that div(fv) = fdiv(v) +v •…

A: We will use simple definition of divergent of vector quantity and gradient of scalar quantity.

Q: 1) Using Laplace transform to solve: y" + 2y - 63y = 2, yco)=1, y'co)=0

A: Step 1: Step 2: Step 3: Step 4:Step 5:Step 6:

Q: d2y/dt2+a2y=b sin wt  ( laplace transform )

A:

Q: Use Raplace Transform to Solue y?+3y'+2y=DUglt) O and %3D guen ylo) =0 y'l6)=1 %3D

A: Use Laplace table to solve this.

Q: y" − 3y' + 2y = 4e²t ; y(0) = -3, y '(0) = 5 -e²t 4e2t et + + (1+t)4e²t - 7et 7 7 7 -e²t4te²t et + 7…

A:

Q: b) f(t) = 3 sin(2t) + 5e³t so

A:

Q: 6. et+7 dt 7. r'e-)* dt 8. I. 4e-" sin 6t dt

A:

Q: a) Using the definition of Laplace transform find L(+²) b) Given the initial value problem 2y" -…

A: (A) To find Lt2 using the Laplace transformation.

Q: Use Laplace transform to solve the following IVP y'(t) — 2y(t) = 8(t – ln(2)), y(0) = 0.

A: Properties of Laplace transform Ly'(t)=sY(s)-y(0)Lδ(t-c)=e-csL-1e-asF(s)=H(t-a)f(t-a)

Q: find the laplace transform of the 1VP: y"'-4y=" e^3+t, y=0=y'(0)=y"(0)=0

A:

Q: Find the Fourier transform of the rectangular pulse: f(t) = 1 -T≤t≤T 2 0 ->t>T

A:

Q: Give the Laplace transform of 5 0Sx and x <I f(x) = -7 sin (2x) 2 +e 2 4 +4 b) O Fe)-+" ( 5 F(s) 7 5…

A:

Q: 2 6. The inverse Laplace transform of F(s) = is s+c a) 2e-c (t+b) u (t+b) b) 2e-c (t-b) u (t-b) c)…

A:

Q: Use Laplace Transform to solue y' +zy y=4et %3D given ylo)"= 2 and y'lo). y'66)=-1 %3D

A:

Q: The inverse Laplace transform of the function F(s) = = a) y(t) = 2 cost + 3 sint b) y(t) = 2e' cost…

A: