Show that the Gaussian solitary wave u(t, x) = e=}(2+31)² is a solution to the logarithmic Korteweg-de Vries equation du + d (u log(1ul) = 0, where (t, x) E R².

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A: Given: hx=∫0x3cosπt2dt.

Q: 1) y = logs Vr² – 1

A: Since you have asked multiple questions in single request.so we will answering only first question.

Q: Consider f (x, y) = log(r² + y²) f.(x, y) = %3D and fy(1,0) = %3D

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Q: (1) Let g(x) :| f(t) dt. What is g'(x)? a

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Q: sin(2t +) dt eu-3 %3D 64

A: Let, I =∫0∞sin2t+π2e2t-3dtApply integration by part ,Ist function = cos(2t) , since sin(π/2+x) =…

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A: Solve the DE using solution from linear and Bernouei's eqn only…

Q: B- Find dy / dx for the following functions: (3 mark) * 1- Y= sin( coshx log, 2 ) 1 Add file 2- (3…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: what is the acceleration at time t=o of a particle if its position is s(t)= cos (3t)-sin(4t

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Q: Chapter 4] If f(x) = dt, where g(x) = 1," (÷) V1+t3 [1 + sin(t²)] dt, find f'

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Q: Find ds/dt when u = 3pai/2 if s = cos u and du/dt = 5.

A: Given,

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Q: ) Find the first derivative: b) y = log7 v1+x

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Q: du 4. (a) If u = log(x³ + x*y + y°), prove that x=+y = 3 du (b) If u = tan-' (x² + 3xy + y²) , prove…

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Q: IfI(u) = [log, (1-2u cos x+ u' )dx, then prove that, I(u)+I(-u)-DI(u²).

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Q: . Find g'(t) if g(t) = sin²(3t² – 1). %3D

A: Given: gt=sin23t2-1

Q: If g(x) = | cos' t dt , then determine g'(x). %3D

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Q: Write the equation „dr te +x dt dr in the form of +p(t)x=q(t). Then identify p(t) and q(t). dt…

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Q: 6. Find the derivative of y = t2 ? = 2t b) = 2t d) = 2 a) 2 dt 4..

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Q: The

A: This is Linear Differential equation in x. We use method of separation of variables.

Q: Let g(x) = [" sin(t²) dt- Calculate g' (3).

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Q: Sin (7€) + Cos (SE) 5지

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Q: Solve the ODE y" + y = et cos(2t) cos(2t) y(0) = 2, y'(0) = 1

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Q: Let F(x) = [ (3t2-3)dt then F"(1) is equal to:

A: The Fundamental Theorem of Calculus : If f is a function that is continuous on an open interval I,…

Q: If you have the equation dx/dt = a,are you allowed to multiply both sides by dt to change the…

A: We have given,