Use Duhamel's formula to solve the following (IVP): in R x (0, 0) Utt – Urz = e cos(r) u(r,0) = 0, u (x,0) = 0 in R

Q: 1 Je* sin( t)dt 1+s* -st Prove that 2

A: We have to prove that:  ∫0∞ e-st sin(t) dt=11+s2

Q: Find '(t), given r(t) = (-4e-²t, 4t² + 3, 3 sin(−3t)). r' (t) =

A: Given vector is as follows

Q: if f(x)=2x-cos(x), Use Newton's method to approximate x, starting from the initial value xo=0

A: Solution:-    fX=2x-cosX           x0=0    f'x=2+sinxwe know      x1=x0-fx0f'x0           =0-f0f'0…

Q: f(x)=cos(x), on [0,2π). solve f(x)=0

A: Given query is to find the Indicated angles .

Q: 1) Solve the IVP: y" + 2y' + 2y = 5 sin t + 5 cos t : y(0) = 1 and y'(0) = 2

A: Given initial value problem is  y''+2y'+2y=5sint+5cost, y(0)=1, y'(0)=2. Let y'=dydt=Dy and…

Q: Solve dN dt = (1¹ 1+ cos(2nt) N where t ≤0 and N(0) = No.

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Q: u(x) = +2x sinh(z/4)− x |, etan''u(t)dt.

A: note :  As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Using the substitution u = x³ – 1, ƒ±₁ x² (x³ — 1)5 dx is equal to which of the following? O O 63 -2…

A: Let, Put

Q: Question 9: Calculate r (t). r (t) = (4t, 2e2t + 1, 2 sin(t)) ®F(t) = (4t, 2e2ª – 1,2 sin(t)) O F(t)…

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Q: Find i (t), given 7(t) = (5 cos(– 2t + 3), 4te', – 2t In( – t)). %3D F (t) =

A: To find the derivative of the given parametric vector, we need to differentiate each component of…

Q: Solve xy (1 + xy) = dy = 1. dx dy dx +x sin 2y =x° cos2 y. %3D · Solve

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Q: Be the equation (2xy cos x-x²y²sinx) dx + 2 x ²³ y c os x d y =0 When solving it by integrating…

A: The given differential equation is ,     2xy2cosx-x2y2sinxdx+2x2ycosx dy =0 Then differential…

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: How to solve this equation y'''-4y''+2y'-8y=6e^(x) - x^(2) + 1 using Methods of Undetermined…

A: Given: Given differential equation is y'''-4y''+2y'-8y=6ex-x2+1 We will solve this equation using…

Q: Given that y(x)= sin( In 2x ). Which of the following formula is CORRECT to find y'(x). v(e) = (In…

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Q: ху — 1 dx x²y 1 xy2 dy = 0

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Q: 2. Find L .2 t2 sin(2@t) F(t) = А. %3D 20 sin(@³r) F(t) = В. 2ot t sin(@t) С. F(t) =- %3D 20 cos(@t)…

A: The given function is Fs=ss2+ω22 We have to find the inverse Laplace transform of F(s). To find this…

Q: Solve the Euler equation d²y dy -x +y= dx dx (In x-1)

A: Solution

Q: udv – v(1 + v cos 2u)du = 0

A: We will find out the required solution.

Q: A yoyo bounces up and down in the vertical direction with its acceleration after tt seconds given by…

A: d^2y/dt^2 = a(t)

Q: 1. / 3 cos³ x sin? x dx

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

Q: Write a in the form a = a-T+ anN at the given value of t without finding T and N. r(t) = (5 e' cos…

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Q: Suppose r' (t) = (cos(t), et, 2t) and r(0) = (1, 0, 3). Find r(t).

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Q: Faraday's law characterizes the voltage drop across an inductor such as Given that di V = L dt 1 di…

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Q: Question 8: The curve C in the ry-plane is parameterized by r (t). Find a parameterization of the…

A: Given:r→'t=4, 4e2t, 2costr→0=0,1,0

Q: y ху" + у' +- %3 (cot(In x)▼

A: Given differential equation is

Q: Determine the solution for this DE 1.) du/dx = (u2-x2) / (x2-u2 ) , u(1) = 0

A:

Q: (b) Calculate e cos(Tx) dx using integration by parts as many times as necessary.

A: Given: ∫excos(πx) dx The formula for integration by parts is given as…

Q: Suppose T(z) = Find a solution to T(x) = | %3! in terms of z. Note: We are using T : C → R? such…

A: The function T is defined from complex plane to cartesian plane. T(a+bi)=a-b0.  Given the line in…

Q: : Mark all the ODEs in the list for which the function y(t) = cos(t) is a solution. A) y' = sin(t)…

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Q: Find the general solution of the DE y" +2 y' +2 y= a cos (wt), where a and w are constants. Note:…

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Q: If z=1+y , I= e' cos(s) and y=e' sin(s), then at (t, s) = (1,0) is as O-2e O 2e O 2e

A: if z=x2+y2 where x=et coss , y=et sins then find ∂z∂s at t,s=1,0

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.