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Finding an Antiderivative In Exercises 53-58, find r( t) that satisfies the initial condition(s).
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Calculus: Early Transcendental Functions
- 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_forwardSolve the case of a vibrating string of length L with initial conditions u(x,0) = f(x), partial u(x,0)/partial x = g(x) and boundary conditions u(0,t) = a(t) and u(L,t) = b(t). Where a(t), b(t) are functions that depend on time. (Hint: Check out Partial Differential Equations: An Introduction by Walter Strauss)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
- Solve the partial differential equation t∂u/∂x−x∂u/∂t=0 with the initial condition u(x,0)=f(x),x>0. Plot the characteristic curves. Note that the critical point is the "o-type" or center in this case.arrow_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_forwardUsing Laplave Transform, evaluate the integro-differential equation y''(x) + 9y(x) = 40ex; y(0) = 5, y'(0)= -2arrow_forward
- Deteremine the area between the curves y= sin(x), y= x^2 + 4, x= -1, and x=2.arrow_forwardUse the Laplace transform to solve the given system of differential equations. (d2x/dt2)+x-y=0 (d2y/dt2)+y-x=0 x(0)=0, x'(0)=-6, y(0)=0, y'(0)=1 x(t)= y(t)=arrow_forwardDetermine the general solution of the system: dx/dt = -x+4y, dy/dt = -x -y Assume we are given the initial conditions x(0)=2 and y(0)=3. Determine the solution that satisfies the given initial conditions. Construct an argument based on the Existence and Uniqueness theorem to explain why this is the only solution.arrow_forward
- Using 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_forwardDraw a free body diagram of the following APPLICATION OF FIRST ORDER LDE (Mixture) 300 gallons of brine solution and 40 pounds of salt are stored in a massive tank. A concentration of 3 lbs/gal is pumped in at a rate of 4 gal/min. At a rate of 3 gal/min, the concentration is pushed out of the tank. After 12 minutes, how much salt is left in the tank?arrow_forward
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