Determine constants a, b, c and d that will produce a quadrature formula f (x) dæ z af (-1)+bf (1) + cf' (–1) + df' (1) -1 that has degree of precision 3.

Q: Let I = f x*cos(x) dx. The approximation of I using the two-point Gaussian quadrature formula is:

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Q: Use the Euler's method to obtain a four-decimal approximation of the indicated value. Use h = 0.1.…

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Q: 1 Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y)…

A: Given: f(x,y)=11-2y-y    at (0,0)

Q: The orthogonal trajectories of r = 5a sin 0 is, where a is an arbitrary constant r = csin 0 O r5 = c…

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Q: 4) Solve the followingIVPusing LaplaceTransform. y′′−6y′+15y=2sin(3t)    ,y(0)=−1,y′(0)=−4

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Q: For the function f(x, y) = (cos x)eY +V 1 + x + y. Use Taylor's formula to find a quadratic…

A: To determine the second degree Taylor quadratic approximation, we need to first find the first…

Q: (a) i. Find Re{sec (x + iy)}.

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Express a general solution to 2x (1 — х)у" + (3 — 10х)у — бу %3D 0 Using Gaussian hypergeometric…

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Q: Q.4 (a) Use the formula S.{f"(z)} = af(0) – a*S.{f(x}} to find Sa{cosh(2r)}. (b) Classify the…

A: Given In part4a:Sαf''x=αf0-α2SαfxIn part4b: yuyy-xuyz+uy=0

Q: Assuming 8x - y +7xy = 0 defines y as a differentiable function of x, use the theorem dx Fx to find…

A: By using partial differentiation .

Q: Find the orthogonal trajectories of r= a sin? 0 where a be the arbitrary constant.

A: We have, r=a sin2θ .....................................................(1) Differentiating w.r.t.…

Q: Show that u(x, y) is harmonic in some domain. u(x, y) = 2x – x³ + 3xy?

A: since Uxx+Uyy=0, then U is harmonic

Q: Compute f cos(Tæ) dæ + 10xy dy, where y is a part of parabola y = 3x? from the point (0,0) to the…

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Q: Show that f(x, y) = ln(x2 + y2) solves Laplace's equation, ∂2z ∂x2  +  ∂2z ∂y2  =…

A: Given Laplace’s equation is

Q: Use the method of separation of variables to solve :=0 in (0,1) × (0,+0), (2) u(0,t) =u(1,t) =0,t…

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Q: 4. Use the I.V.T and Rolle's Theorem/M.V.T to prove that the equation below has a unique solution in…

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Q: Assume that a spring does not follow Hooke’s Law. Instead, the force required to stretch the spring…

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Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: Solve by theorems of linearity and translation. f(t) = cos t + sin t

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Q: For the functions w = 8x + 3y, x= cos t, and y = sin t, express dw as a function of t, both by using…

A: The given function is w=8x2+3y2, x=cost and y=sint. We have to express dwdt as a function of t.…

Q: a) Exaluate: %3D :(?)6 puo (x puif (9 and (3 +x x <O %3D for which f (x)=9). Give exact working out…

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Q: (1) Vf (f is a scalar function) 2) ř(t) and r(s), t is time and s is arch length.

A: This is a problem of vector calculus.

Q: Show that the Euler-Cauchy Equation ry" + ary + by = 0, r+0 can be transformed into the constant…

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Q: (b) Use Leibniz's product rule to derive the quotient rule y du- v dy d YY

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Yn + 1 = Yn +…

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Q: (a) Assume that traffic flows at a rate of 240 vehicles per hour on the stretch of road between…

A: The number of vehicles coming and leaving the signal is same in all the signals. That is, Total…

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y)…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: (a) Consider the equation e'ux+2ayuy +2e*uy-sin(y) u =0, where a is a real constant. Determine the…

A: We will use basic knowledge of PDEs to answer this question.

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: EXERCISE 6. dy +y dr where A is an arbitrary constant () - (+*) + A, Given that prove that tan

A: Solve By substitution

Q: Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the…

A: The objective is to evaluate ∫CF·dr where Fx,y=2y-3cosxi+4x+6sinyj and C goes counterclockwise…

Q: The orthogonal trajectories of r = 5a sin 0 is , where a is an arbitrary constant p = c cos 0 r=…

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Q: In the figure, ZY = ZX and YZ is produced to T. Prove that YT>XT. Y T

A: Given a diagram: ZY=ZX To  prove YT&gt;XT

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: Given differential equation.y' = x+y2, y(0)=0,.we have to find y(0.2) .we use Euler's method…

Q: Show that u (x, y) is harmonic in some domain and |find a harmonic conjugate v(x, y) when y (a) u(x,…

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Q: =y一;-T十;-一

A: Draw the family of these one parameter DE's of bifurcation.    Further, to solve these.

Q: Exercise 6 Using the method of variation of parameters, the particular solu- tion y, for the DE y" +…

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Q: the orthogonal trajectories of r = a sin? 0 where a be the arbitrary constant

A: Orthogonal trajectory

Q: Q5: Find W,e if W = f(x,y), xr cos6 and y =r sino, then prove that r Wre =-yf + xfy+ xy(fyy-fex) +…

A: The given problem is to find the double partial derivative of the given multi variable function and…

Q: Express in form y=f(x) by eleminating the parameter x = e−2t , y = 6e4t

A: Here, x=e-2t and y=6e4t

Q: If a and ß are positive constants, show that the particular solution of dy %3D У (В — ау) dt that…

A: Given that dydt=yβ-αy and y=y0 at t=0 We have to find the solution of the above equation.

Q: Show that the function f(x,y) =x² cos² y+ ysin² x+xe" , satisfies the 2 COS Lipschitz condition in…

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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.