2.99 See Answer

Question: Solve the differential equation using the method


Solve the differential equation using the method of variation of parameters.
y'' + y = sec3x, 0 < x < π/2


> (a). Evaluate the line integral&acirc;&#136;&laquo;C F &acirc;&#136;&#153; dr, where F (x, y, z) = x i - z j + y k and C is given by r(t) = 2t i + 3t j - t2 k, -1 (b). Illustrate part (a) by using a computer to graph C and the vectors from the vector fie

> Evaluate the line integral, where C is the given curve. ∫C xy4 ds, C is the right half of the circle x2 + y2 = 16

> Use a calculator to evaluate the line integral correct to four decimal places. ∫C z ln (x + y) ds, where C has parametric equations x = 1 + 3t, y = 2 + t2, z = t4, -1 < t < 1

> Find &acirc;&#136;&laquo;&acirc;&#136;&laquo;S F &acirc;&#136;&#153; n dS, where F (x, y, z) = x i + y j + z k and S is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed). ZA (0, 2, 2)

> Use a calculator to evaluate the line integral correct to four decimal places. ∫C xy arctan z ds, where C has parametric equations x = t2, y = t3, z = √t, 1 < t < 2

> (a). Find a parametric representation for the torus obtained by rotating about the z-axis the circle in the xz-plane with center (b, 0, 0) and radius a (b). Use the parametric equations found in part (a) to graph the torus for several values of a and b.

> Use a calculator to evaluate the line integral correct to four decimal places. ∫C F ∙ dr, where F (x, y) = √(x + y) i + (y/x) j and r(t) = sin2 t i + sin t cos t j, π/6 < t < π/3

> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y, z) = x i + y j + xy k, r(t) = cos t i + sin t j + t k, 0 < t < π

> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y, z) = sin x i + cos y j + xz k, r (t) = t3 i - t2 j + t k, 0 < t < 1

> (a). Show that the parametric equations x = a cosh u cos v, y = b cosh u sin v, z = c sinh u, represent a hyperboloid of one sheet. (b). Use the parametric equations in part (a) to graph the hyperboloid for the case a = 1, b = 2, c = 3. (c). Set up, but

> Evaluate the line integral, where C is the given curve. ∫C (x/y) ds, C: x = t 3, y = t 4, 1 < t < 2

> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y) = xy2 i - x2 j, r(t) = t3 i + t2 j, 0 < t < 1

> Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 < x < 4, 0 < y < 1.

> Find the area of the surface with vector equation r (u, v) = 〈cos3u cos3v, sin3u cos3v, sin3v〉, 0 < u < u, 0 < v < 2π. State your answer corrects to four decimal places.

> Let F (x, y) = (2x3 + 2xy2 - 2y) i + (2y3 + 2x2y + 2x) j x2 + y2 Evaluate &acirc;&#136;&laquo;C F &acirc;&#136;&#153; dr, where C is shown in the figure. C

> (a). Use the Midpoint Rule for double integrals (see Section 15.1) with six squares to estimate the area of the surface z = 1/ (1 + x2 + y2), 0 < x < 6, 0 < y < 4. (b). Use a computer algebra system to approximate the surface area in part (a) to four dec

> Find, to four decimal places, the area of the part of the surface z = (1 + x2)/ (1 + y2) that lies above the square |x | + |y | < 1. Illustrate by graphing this part of the surface.

> Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = ln (x2 + y2 + 2) that lies above the disk x2 + y2 < 1

> Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = cos (x2 + y2) that lies inside the cylinder x2 + y2 = 1

> If the equation of a surface S is z = f (x, y), where x2 + y2 < R2, and you know that | fx | < 1 and | fy | < 1, what can you say about A (S)?

> If C is a smooth curve given by a vector function r (t), a r• dr = {[Ir(b)|? – \r(a)F]

> Let F be an inverse square field, that is, F (r) = cr/|r |3 for some constant c, where r = x i + y j + z k. Show that the flux of F across a sphere S with center the origin is independent of the radius of S.

> The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of radius a with center at the center of the ball.

> The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u (x, y, z) = 2y2 + 2z2. Find the rate of heat flow inward across the cylindrical surface y2 + z2 = 6, 0 < x < 4.

> Use Gauss’s Law to find the charge enclosed by the cube with vertices (±1, ±1, ±1) if the electric field is E (x, y, z) = x i + y j + z k

> Let F (x, y, z) = (3x2yz - 3y) i + (x3z - 3x) j + (x3y + 2z) k Evaluate &acirc;&#136;&laquo;C F &acirc;&#136;&#153; dr, where C is the curve with initial point (0, 0, 2) and terminal point (0, 3, 0) shown in the figure. ZA (0, 0, 2) (0, 3, 0) (1, 1,

> Use Gauss’s Law to find the charge contained in the solid hemisphere x2 + y2 + z2 < a2, z > 0, if the electric field is E (x, y, z) = x i + y j + 2z k

> Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x2 + y2 + z2 = 9, z > 0.

> A fluid has density 870 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 4, 0 < z < 1.

> Find the area of the surface. The part of the cone z = √x2 + y2 that lies between the plane y = x and the cylinder y = x2

> (a). Give an integral expression for the moment of inertia Iz about the z-axis of a thin sheet in the shape of a surface S if the density function is ρ. (b). Find the moment of inertia about the z-axis of the funnel in Exercise 40 Exercise 40: Find the

> Find the mass of a thin funnel in the shape of a cone z = √x2 + y2, 1 < z < 4, if its density function is ρ (x, y, z) = 10 - z.

> Find the center of mass of the hemisphere x2 + y2 + z2 = a2, z > 0, if it has constant density.

> Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. r(u, v) = (1 – u² – v²) i – vj – u k; (-1, –1, –1)

> Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. r(u, v) = ưi + 2u sin vj + u cos vk; u = 1, v =0

> Find the flux of F (x, y, z) = sin (xyz) i + x2y j + z2ex/5 k across the part of the cylinder 4y2 + z2 = 4 that lies above the xy-plane and between the planes x = -2 and x = 2 with upward orientation. Illustrate by using a computer algebra system to draw

> Compute the outward flux of F (x, y, z) = x i + y j + z k/ (x2 + y2 + z2)3/2 through the ellipsoid 4x2 + 9y2 + 6z2 = 36.

> Find the value of ∫∫S x2y2z2 dS correct to four decimal places, where S is the part of the paraboloid z = 3 - 2x2 - y2 that lies above the xy-plane.

> Find the exact value of ∫∫S xyz dS, where S is the surface z = x2y2, 0 < x < 1, 0 < y < 2.

> Evaluate ∫∫S (x2 + y2 + z2) dS correct to four decimal places, where S is the surface z = xey, 0 < x < 1, 0 < y < 1.

> The surface with parametric equations x = 2 cos θ + r cos (θ /2) y = 2 sin θ + r cos (θ /2) z = r sins (θ /2) where -1/2 < r < 12 and 0

> Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that These surface and triple integrals of vector functions are vectors defined by integrating each component functio

> Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. (SVg – gVf) •n ds = [[[ (fV²g – gV²f) aV E

> Solve the boundary-value problem, if possible. y'' = y', y (0) = 1, y (1) = 2

> Solve the differential equation using the method of variation of parameters. y'' + 4y' + 4y = e-2x/x3

> Solve the differential equation using the method of variation of parameters. y'' – y' + y = ex/1 + x2

> Solve the differential equation using the method of variation of parameters. y'' + 3y' + 2y = sin (ex)

> Verify that the Divergence Theorem is true for the vector field F (x, y, z) = x i + y j + z k, where E is the unit ball x2 + y2 + z2 < 1.

> Solve the differential equation using the method of variation of parameters. y'' - 3y' + 2y = 1/1 + e-x

> Solve the differential equation using the method of variation of parameters. y'' + y = sec2x, 0 < x < π/2

> Plot the vector field and guess where div F &gt; 0 and where div F F(x, y) = (x², y²)

> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = zexy i - 3zexy j + xy k, S is the p

> Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. у" — 2у' — Зу %3х+2 3y = x + 2

> Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters. 4y" + у %3 сos x

> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y" + 4y = ex + x sin 2.x

> Verify that the solution to Equation 1 can be written in the form x (t) = A cos (ω t + δ).

> The battery in Exercise 14 is replaced by a generator producing a voltage of E (t) = 12 sin 10t. Exercise 14: A series circuit contains a resistor with R = 24 V, an inductor with L = 2 H, a capacitor with C = 0.005 F, and a 12-V battery. (a). Find the

> Use the Divergence Theorem to calculate the surface integral ∫∫S F ∙ dS, where F (x, y, z) = x3 i + y3 j + z3 k and S is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the planes z = 0 and z = 2.

> The battery in Exercise 13 is replaced by a generator producing a voltage of E (t) = 12 sin 10t. Find the charge at time t. Exercise 13: A series circuit consists of a resistor with R = 20 V, an inductor with L = 1 H, a capacitor with C = 0.002 F, and

> A series circuit contains a resistor with R = 24 V, an inductor with L = 2 H, a capacitor with C = 0.005 F, and a 12-V battery. The initial charge is Q = 0.001 C and the initial current is 0. (a). Find the charge and current at time t. (b). Graph the cha

> A series circuit consists of a resistor with R = 20 V, an inductor with L = 1 H, a capacitor with C = 0.002 F, and a 12-V battery. If the initial charge and current are both 0, find the charge and current at time t.

> The solution of the initial-value problem x2y'' + xy' + x2y = 0 y (0) = 1 y' (0) = 0 is called a Bessel function of order 0. (a). Solve the initial-value problem to find a power series expansion for the Bessel function. (b). Graph several Taylor polynomi

> Use power series to solve the differential equation. y'' + x2y' + xy = 0, y (0) = 0, y' (0) = 1

> Use power series to solve the differential equation. y'' + x2y = 0, y (0) = 1, y'(0) = 0

> Use power series to solve the differential equation. y'' – xy' - y = 0, y (0) = 1, y' (0) = 0

> Use power series to solve the differential equation. y'' = xy

> Use power series to solve the differential equation. (x – 1) y' + y' = 0

> Use power series to solve the differential equation. y'' = y

> Use Stokes’ Theorem to evaluate ∫C F ∙ dr, where F (x, y, z) = xy i + yz j + z x k, and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed from above.

> Use power series to solve the differential equation. y'' + xy' + y = 0

> Use power series to solve the differential equation. (x – 3) y' + 2y = 0

> Use power series to solve the differential equation. y' = x2y

> Solve the differential equation. Y'' - 6y' + 9y = 0

> Use power series to solve the differential equation. y' – y = 0

> How do you find the cross product a × b of two vectors if you know their lengths and the angle between them? What if you know their components?

> Write expressions for the scalar and vector projections of b onto a. Illustrate with diagrams.

> How do you find the dot product a ∙ b of two vectors if you know their lengths and the angle between them? What if you know their components?

> How do you find the vector from one point to another?

> How do you add two vectors geometrically? How do you add them algebraically?

> Use Stokes’ Theorem to evaluate ∫∫S curl F ∙ dS, where F (x, y, z) = x2yz i + yz2 j + z3exy k, S is the part of the sphere x2 + y2 + z2 = 5 that lies above the plane z = 1, and S is oriented upward.

> Write equations in standard form of the six types of quadric surfaces.

> How do you find the angle between two intersecting planes?

> (a). How do you find the area of the parallelogram determined by a and b? (b). How do you find the volume of the parallelepiped determined by a, b, and c?

> What is the difference between a vector and a scalar?

> Solve the differential equation. y'' - 4y' + 13y = 0

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. y" – y = xe2", y(0) = 0, y'(0) = 1

> Solve the differential equation. 3y'' = 4y'

> For the spring in Exercise 4, find the damping constant that would produce critical damping. Exercise 4: A force of 13 N is needed to keep a spring with a 2-kg mass stretched 0.25 m beyond its natural length. The damping constant of the spring is c = 8

> For the spring in Exercise 3, find the mass that would produce critical damping. Exercise 3: A spring with a mass of 2 kg has damping constant 14, and a force of 6 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring i

> A force of 13 N is needed to keep a spring with a 2-kg mass stretched 0.25 m beyond its natural length. The damping constant of the spring is c = 8. (a). If the mass starts at the equilibrium position with a velocity of 0.5 m/s, find its position at time

> Verify that Stokes’ Theorem is true for the vector field F (x, y, z) = x2 i + y2 j + z2 k, where S is the part of the paraboloid z = 1 - x2 - y2 that lies above the xy-plane and S has upward orientation.

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. 9y" + y = e2*

> Consider a spring subject to a frictional or damping force. (a). In the critically damped case, the motion is given by x = c1ert + c2tert. Show that the graph of x crosses the t-axis whenever c1 and c2 have opposite signs. (b). In the overdamped case, th

> Solve the differential equation. 2 d2y/dt2 + 2 dy/dt - y = 0

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. y" + y' – 2y = x + sin 2x, y(0) = 1, y'(0) = 0 %3D

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. y'' + 2y' - 8y = 1 - 2x2

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. у" — 4y' + 4у %3х — sin x

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. у" — 4у' + 5у — е*

> Solve the differential equation or initial-value problem using the method of undetermined coefficients. у" — 2у' + 2у х+e*

> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. у" + 2y' + 10у 3D х?е-* сos 3x

2.99

See Answer