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sql server reporting services barcode font Plot the vector field F(x, y) = y i + x j . in Software
EXAMPLE 9 Plot the vector field F(x, y) = y i + x j . Reading QR Code In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Draw QR In None Using Barcode encoder for Software Control to generate, create Denso QR Bar Code image in Software applications. VectorPlot [{ y, x}, {x, 5, 5}, {y, 5, 5}] QR Code Reader In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Generate QR Code In C# Using Barcode generator for Visual Studio .NET Control to generate, create Denso QR Bar Code image in .NET applications. Any firstorder differential equation can be used to define a vector field. Indeed, the vector field i + f(x, y) j, corresponding to the equation dy = f ( x , y) , generates a field whose vectors are tangent to the solution. The Generating QR Code ISO/IEC18004 In VS .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications. Encoding QR Code In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. next example, although simple, illustrates this nicely.
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DataMatrix Generation In None Using Barcode encoder for Excel Control to generate, create Data Matrix image in Microsoft Excel applications. Decode GS1  13 In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. dy = 2 x + y. Then the soludx
tions with initial conditions y(0) = 2, 1, 0, 1, and 2 are plotted on the vector field for comparison. vf = VectorPlot[{1, 2 x + y}, {x, 2, 1}, {y, 4, 6}, Axes Automatic, VectorScale Small] 4 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Ordinary Differential Equations
solutions = Table[DSolve[{y'[x] 2 x + y[x], y[0] k}, y[x], x], {k, 2, 2}]; Do[g[k]= Plot[solutions[[k, 1, 1, 2]], {x, 2, 1}, PlotRange All, Frame True, PlotStyle Thickness[.005]], {k, 1, 5}] Show[g[1], g[2], g[3], g[4], g[5], vf, AspectRatio 1, PlotRange { 4,6}] 4 2.0 1.5 1.0 0.5 A system of differential equations consists of n differential equations involving n + 1 variables. Solving a system of differential equations with Mathematica is similar to solving a single equation. This example illustrates how to solve the system dx = t 2 , dy = t 3 with initial conditions dt dt x(0) = 2, y(0) = 3. The equation and its initial conditions are contained within a list. EXAMPLE 12
solution = DSolve[{x'[t] t2, y'[t] t3, x[0] 2, y[0] 3}, {x[t], y[t]}, t] x[t] 1 (6 + t3 ), y[t] 1 (12 + t4 ) 3 4 Instead of specifying the values of f and its derivatives at a single point, values at two distinct points may be given. The problem of solving the differential equation then becomes known as a boundary value problem. However, unlike initial value problems, which can be shown to have unique solutions for a wide variety of cases, a boundary value problem may have no solution even for the simplest of equations. EXAMPLE 13 The equation
d2y + y = 0 with boundary conditions y(0) = 0, y( ) = 1 has no solution. dx 2 DSolve[{y''[x]+ y[x] 0, y[0] 0, y[o] 1}, y[x], x] DSolve bvnul : For some branches of the general solution, the given boundary conditions lead to an empty solution. {} The same equation with y(0) = 0, y( /2) = 1 has a unique solution. DSolve[{y''[x]+ y[x] 0, y[0] 0, y[o/2] 1}, y[x], x] {{y[x] Sin[x]}} Ordinary Differential Equations
SOLVED PROBLEMS
11.1 Solve the differential equation dy = x y with initial condition y(1) = 2 and graph the solution for 2 x 2. SOLUTION
solution = DSolve[{y'[x] x y[x], y[1] 2}, y[x], x] {{y[x] 2 1 + x2 2 2
8 6 4 2 Plot[y[x] /.solution, {x, 2, 2}] 11.2 Plot the vector field for the differential equation of Problem 11.1.
SOLUTION
VectorPlot[{1, x y}, {x, 2, 2}, {y, 10, 10}, VectorScale Tiny, Axes Automatic] 10 2 1 0 1 2 11.3 Plot the vector field for the equation dy = x 2 + y together with its solutions for y(0) = 0, 1, 2, 3, and 4. SOLUTION
vf = VectorPlot [{1, x2 + y}, {x, 0, 1}, {y, 0, 12}, Axes Automatic, VectorScale Tiny]; Ordinary Differential Equations
solutions = Table[DSolve[{y'[x] x2 + y[x], y[0] k}, y[x], x], {k, 0, 4}]; Do[g[k]= Plot[solutions[[k, 1, 1, 2]], {x, 0, 1}, PlotStyle Thickness[.007], PlotRange All], {k, 1, 5}] Show[g[1], g[2], g[3], g[4], g[5], vf, Frame True, AspectRatio 1] 0 0.0 0.2 0.4 0.6 0.8 1.0 11.4 The escape velocity is the minimum velocity with which an object must be propelled in order to escape the gravitational field of a celestial body. Compute the escape velocity for the planet Earth.

