answer:Your derived solution is correct: y(x) = c_1e^{x} + c_2e^{-x} + left(frac{1}{2}right)xsinh(x) The initial condition y(0) = 2 is used to find the constants c_1 and c_2. At x = 0, we have: y(0) = c_1e^{0} + c_2e^{-0} + left(frac{1}{2}right)0sinh(0) = c_1 + c_2 = 2 Additionally, differentiate y(x) to find y'(x): y'(x) = c_1e^{x} - c_2e^{-x} + left(frac{1}{2}right)sinh(x) + left(frac{1}{2}right)xcosh(x) Then, apply the second initial condition y'(0) = 1 (there was a typo in the original answer, it should be 1, not 12): y'(0) = c_1e^{0} - c_2e^{-0} + left(frac{1}{2}right)sinh(0) + left(frac{1}{2}right)0cosh(0) = c_1 - c_2 = 1 Now, solve the system of equations: begin{cases} c_1 + c_2 = 2 c_1 - c_2 = 1 end{cases} Adding these equations, we get 2c_1 = 3, so c_1 = frac{3}{2}. Substituting this into the first equation, we find c_2 = 2 - c_1 = 2 - frac{3}{2} = frac{1}{2}. The particular solution to the differential equation with the given initial conditions is: y(x) = frac{3}{2}e^{x} + frac{1}{2}e^{-x} + left(frac{1}{2}right)xsinh(x)

answer:Using the Divergence Theorem, we have: {eq}displaystyle int int _S Fcdot dS = int int int div(vec F) dV {/eq} Calculating the divergence of {eq}vec F {/eq}: {eq}displaystyle div(vec F) = frac{partial}{partial x}(x^2) + frac{partial}{partial y}(y^2) + frac{partial}{partial z}(z^2) = 2x + 2y + 2z {/eq} The limits of integration are: {eq}0 leq rho leq 2, quad 0 leq theta leq 2pi, quad rho^2 leq z leq 4 {/eq} where {eq}rho {/eq} is the radial distance in cylindrical coordinates. Therefore, the integral becomes: {eq}displaystyle int int _S Fcdot dS = int_{0}^{2pi} int_{0}^{2} int_{rho^2}^{4} (2rhocostheta + 2rhosintheta + 2z) rho dz drho dtheta {/eq} Evaluating the integral, we get: {eq}displaystyle int int _S Fcdot dS = boxed{frac{128pi}{3}} {/eq}

answer:Depreciation expense per year = (Cost of asset - Salvage value) / Useful life Depreciation expense per year = (15,000 - 2,000) / 5 Depreciation expense per year = 2,600

answer:Given a smooth, continuously differentiable ({eq}C^1{/eq}) vector field {eq}k{/eq} defined on a smooth manifold {eq}N{/eq}, the flow of this vector field can be defined as {eq}boxed{phi_{tk}(p) = gamma_p(t)}{/eq}. Here, {eq}gamma_p{/eq} represents an integral curve satisfying the following conditions: 1. {eq}gamma_p(0) = p{/eq}, ensuring that the curve passes through point {eq}p{/eq} at time {eq}t = 0{/eq}. 2. The curve is determined by the vector field {eq}k{/eq}, meaning that the derivative of {eq}gamma_p{/eq} with respect to time {eq}t{/eq} is equal to the value of the vector field {eq}k{/eq} at the point {eq}gamma_p(t){/eq}. This flow defines a one-parameter family of diffeomorphisms that describe the movement of points on the manifold under the influence of the vector field.