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k Log 12 5 Scan QR Code JIS X 0510 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QR Code Printer In None Using Barcode creation for Software Control to generate, create QR Code image in Software applications. population[4] /. k Log[12/5] 6912 Decoding QR Code JIS X 0510 In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. QR Code Encoder In Visual C# Using Barcode encoder for VS .NET Control to generate, create QR Code 2d barcode image in .NET applications. 11.10 The equation governing the amount of current, I, flowing through a simple resistance inductance dI circuit when an EMF (voltage) E is applied is L + RI = E. The units for E, I, and L are, dt respectively, volts, amperes, and henries. If R = 10 ohms, L = 1 henry, the EMF source is an alternating voltage whose equation is E(t) = 10 sin 5t, and the current is initially 4 amperes, find an expression for the current at time t and plot the graph of the current for the first 3 seconds. Printing QRCode In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Encode QR Code JIS X 0510 In .NET Framework Using Barcode maker for Visual Studio .NET Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications. SOLUTION
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Make Data Matrix In VB.NET Using Barcode drawer for .NET framework Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications. Decode Barcode In .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. 11.11 If a spring with mass m attached at one end is suspended from its other end, it will come to rest in an equilibrium position. If the system is then perturbed by releasing the mass with an initial velocity of v0 at a distance y0 below its equilibrium position, its motion satisfies the differential equation d2y dy m 2 + a + k y = 0, y '(0) = v0 , y(0) = y0 . a is a damping constant (determined experimentally) dt dt due to friction and air resistance, and k is the spring constant given in Hooke s law. A mass of slug is attached to a spring with a spring constant, k, of 6 lb/ft. The mass is pulled downward from its equilibrium position 1 ft and then released. Assuming a damping constant, a, of , determine the motion of the mass and sketch its graph for the first 5 seconds. SOLUTION
m = 1/4; y0 = 1; v0 = 0; a = 1/2; k = 6; solution = DSolve[{m y''[t]+ a y'[t]+ k y[t] 0, y'[0] v0, y[0] y0}, y[t], t] y[t] 1 t 23 Cos[ 23 t] + 23 Sin [ 23 t] 23 height[t_]= solution[[1, 1, 2]]; Plot[height[t], {t, 0, 5}, AxesLabel {"t","Height"}] Height 0.5
0.5 1.0 11.12 If a cable of uniform crosssection is suspended between two supports, the cable will sag forming a curve called a catenary. If we assume the lowest point on the curve to lie on the yaxis, a distance y0 above the origin, the differential equation governing its shape can be shown to be d2y 1 dy y(0) = y , y '(0) = 0 , where a is a positive constant dependent upon the = 1+ 0 dx dx 2 a physical properties of the cable. Find an equation of the catenary and sketch its graph.
Ordinary Differential Equations
SOLUTION
2 solution = DSolve y''[x] 1 1 + y'[x] , y'[0] 0, y[0] y0 , y[x], x a Solve ifun : Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. y[x] a + y0 a Cosh x , y[x] a + y0 + a Cosh x a a
Since = y0 > 0, the second solution applies. We take y0 = a = 1 and plot its graph. catenary[x_]= solution[[2, 1, 2]] /. {a 1, y0 1}; Plot[catenary[x], {x, 1, 1},Ticks {Automatic, {0, 1, 2}}, PlotRange {0, 2}] 1.0 0.5 11.13 The logistic equation for population growth, dp = ap bp2 , was discovered in the midnineteenth dt century by the biologist Pierre Verhulst. The constant b is generally small in comparison to a so that for small population size p the quadratic term in p will be negligible and the population will grow approximately exponentially. For large p, however, the quadratic term serves to slow down the rate of growth of the population. Solve the logistic equation and sketch the solution for a = 2, b = .05, and an initial population p0 = 1 (thousand). Then determine the limiting value of the population as t .

