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Finding an Antiderivative In Exercises 53-58, find r(t) that satisfies the initial condition(s).
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Chapter 12 Solutions
Calculus
- Showing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,ebx}, abarrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forwardUsing fourier transform, obtain the solution of the heat equation in integral form satisfying the initial condition u(x,0) = f(x) where f(x) = (sinx)/x.arrow_forward
- Using Laplave Transform, evaluate the integro-differential equation y''(x) + 9y(x) = 40ex; y(0) = 5, y'(0)= -2arrow_forwardUsing a Differential as an Approximation, f(x, y) = 16 − x2 − y2 find f (2, 1) and f (2.1, 1.05) and calculate ∆z, and use the total differential dz to approximate ∆z.arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = (-5x + 2y) i + (6x - 9y) j; C is the region bounded above by y = -5x 2 + 250 and below by y=5x2 in the first quadrantarrow_forward
- Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forwardA. Transform the equation to its equivalent linear equation in standard form and identify P(x) and Q(x) B. Determine the integrating factor. C. Find the integral for ∫ Q(x) μ(x) dx D. Find the general solutionarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning