Use power series to solve the differential equation. y'' + xy' + y = 0
> 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 ∫C F ∙ 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 = sec3x, 0 < x < π/2
> 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 > 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. (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
> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y" + 3y' – 4y = (x³ + x)e*
> Evaluate the surface integral. ∫∫S F ∙ dS, where F (x, y, z) = x2 i + xy j + z k and S is the part of the paraboloid z = x2 + y2 below the plane z = 1 with upward orientation
> (a). Write the general form of a second-order nonhomogeneous linear differential equation with constant coefficients. (b). What is the complementary equation? How does it help solve the original differential equation? (c). Explain how the method of undet
> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. у" — Зу' + 2у — е* + sin x
> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y" + 4y = cos 4x + cos 2x %3D COS
> Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y" – y' – 2y = xe*cos x
> Solve the differential equation. d2R /dt2 + 6 dR/dt + 34R = 0
> Solve the differential equation. y = y''
> Solve the differential equation. 9y'' + 4y = 0
> Solve the differential equation. 4y'' + 4y' + y = 0
> Solve the differential equation. y'' + y' - 12y = 0
> Solve the differential equation. y'' + 2y = 0
> 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. Sſ Da f ds = {[[ v?fav v²fdV E
> Evaluate the surface integral. ∫∫S F ∙ dS, where F (x, y, z) = xz i - 2y j + 3x k and S is the sphere x2 + y2 + z2 = 4 with outward orientation
> 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. curl F· dS = 0
> Solve the boundary-value problem, if possible. y'' + 6y' = 0, y (0) = 1, y (1) = 0
> 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. || a·n ds = 0, where a is a constant vector %3
> Solve the initial-value problem. 4y'' + 4y' + 3y = 0, y (0) = 0, y' (0) = 1
> Solve the initial-value problem. y'' – y' - 12y = 0, y (1) = 0, y' (1) = 1
> Solve the initial-value problem. 3y'' - 2y' - y = 0, y (0) = 0, y' (0) = -4
> Solve the initial-value problem. 9y'' + 12y' + 4y = 0, y (0) = 1, y' (0) = 0
