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In Problems 1 through 10, find a function
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Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis
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- Problem 3 In class, we solved for the vorticity distribution for a "real" line vortex diffusing in a viscous fluid. Integrate this vorticity distribution to find the tangential velocity as a function of radius. Plot the velocity distributions for a a line vortex of circulation 0.5 mls in 20 Β°C air for times of 1, 10, and 100 seconds.arrow_forwardSuppose that a parachutist with linear drag (m=50 kg, c=12.5kg/s) jumps from an airplane flying at an altitude of a kilometer with a horizontal velocity of 220 m/s relative to the ground. a) Write a system of four differential equations for x,y,vx=dx/dt and vy=dy/dt. b) If theinitial horizontal position is defined as x=0, use Eulerβs methods with οt=0.4 s to compute the jumperβs position over the first 40 s. c) Develop plots of y versus t and y versus x. Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open.arrow_forwardfind the general solution to the following cauchy-euler differential equation. x2y''+xy'-9y=2xInxarrow_forward
- Solve the following system of linear equation by (a) Gaussian elimination (b) Gauss Jordan and (c) Inverse matrix. Show complete solution. (Manual Computation)arrow_forwardQ2/ The pipe in Fig. is driven by pressurized air in the tank. What is the friction factor (f) when the water flow rate through pipe is ( 85 m/hr ) and the pressure at point 1 is (2500 kPa). (25Marks) 30m smooth pipe d = 70mm open jet P1 1 90m 15m 60marrow_forwardSolve the following equations. Be sure to check the potential solution(s) in the original equation, to see whether it (they) are in the domain. (a) log, (r? βx β 2) = 2arrow_forward
- 1. Determine the equation of the line through the given point (a) parallel and (b) perpendicular to the given lineΒ Given: Point: ( -1,-4) Line: 4x-2y=3arrow_forward2. Heat conduction in a square plate Three sides of a rectangular plate (@ = 5 m, b = 4 m) are kept at a temperature of 0 C and one side is kept at a temperature C, as shown in the figure. Determine and plot the ; temperature distribution T(x, y) in the plate. The temperature distribution, T(x, y) in the plate can be determined by solving the two-dimensional heat equation. For the given boundary conditions T(x, y) can be expressed analytically by a Fourier series (Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, 1993):arrow_forwardSITUATION 1 (Fluid Flow in a Closed Conduit) Consider a fluid, with density (p) of 998.21 kg/mΒ³ and dynamic viscosity (u) of 1.002 x 103 N-s/mΒ², flowing in a 2000-meter long, 50-mm diameter smooth round pipe with velocity of 2.5 m/s. The energy loss on the pipe flow (he) due to friction between the pipe and the fluid is determined using Darcy-Weisbach equation, given as hβ = f (Β²) (1/1) where f is the friction factor, L is the length of the pipe, D is the diameter of the pipe, V is the velocity of the flow, and g is the gravitational acceleration. The friction factor may be determined using an empirical equation developed by Nikuradse for flow in smooth pipes, given as 1 =0.869 In (Reβ7)-0.8 where Re is the Reynolds number of the flow, determined as VDp Rβ = ΞΌl The friction factor equation given is only valid for flows with Reynolds number higher than 4000 (turbulent flow). Guide Questions: Determine the Reynolds number of the flow. Is the Nikuradse equation for friction factorβ¦arrow_forward
- Find A PARTICULAR solution of the following differential equation by using UNDE-TERMINED COEFFICIENTS METHOD. yβ + 3yβ + 2y = ex sin xarrow_forwardA rope of negligible mass is wrapped around a 225-kg solid cylinder of radius 0.400 m. The cylinder is suspended several meters off the ground with its axis oriented horizontally, and turns on that axis without friction. (a) If a 75.0-kg man takes hold of the free end of the rope and falls under the force of gravity, what is his acceleration? m/sΒ² (b) What is the angular acceleration of the cylinder? rad/sΒ² (c) If the mass of the rope were not neglected, what would happen to the angular acceleration of the cylinder as the man falls?arrow_forwardIn Problems 1-24, find the general solution of the given differ- ential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. 8. y' 2y + xΒ² + 5arrow_forward
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