Ode45 solver

Fourth, you can inspect the numerical value of y at a given t. See Table 7.4 for ode45 results. Notice, however, that the MATLAB ode45 solver used Af = 0.0125 not Af = 0.1. 

Table 7.4 The solution for dt using the MATLAB ode45 solver and the anal54ical.

Figure 7.2 Plot of y vs. t as a solution for dy/dt =y using MATLAB ode45 solver with At = 0.0125.

Call the MATLAB ode45 solver via the myodein.m file ... 

Second, create callmultiode.m (a script M-file) that contains the initial value for eachy the lower and upper limit for integration the options attributes, which is a structure of optional parameters that change the default integration properties a call to the MATLAB ode45 solver and a plot command fory as a function of t. Figure 7.7 embodies such a list of commands. 

Figure 7.7 callmultiode.m (script M-file) that defines integration-related parameters and calls the MATLAB ode45 solver using a function handle (i.e., multiode) to multiode function. 

Figure 7.8 The solution for a set of three ODEs using the MATLAB ode45 solver.

Let us numerically solve the aforementioned ode using the ode45 solver. 

Figure 7.10 shows the callmultiode7 8.m file that calls upon MATLAB s ode45 solver and makes a plot of the variables y s vs. t. 

Figure 7.11 The plot of y2> andy na/ as a function of t. Notice howyi exactly matches withy, [Eg. t7.20i1. yi andy2 were obtained via ode45 solver. 

Let us numerically solve the aforementioned ode using the MATLAB ode45 solver. First, the second-order ODE will be transformed into a set of two first-order ODEs via the following transformation ... 

Use the MATLAB ode45 solver to solve Problem 7.3. First, define the derivative using the function M-file named Pr7 4.m. Second, write the calling code (main code) that defines the initial value of y, the time interval, calls the MATLAB ode45 solver, defines the exact... 

Solve numerically (using MATLAB ode45 solver) the following second-order ordinary differential equation, making, first, the proper transformation into a set of two first-order ODEs y"+5y = f. 

Solve numerically (using MATLAB ode45 solver) the first-order differential equation ... 

Another method that uses fourth- and fifth-order embedded pairs is the Dormand-Prince method. The Dormand-Prince method is more accurate than the Runge-Kutta Fehlberg method and it is used by the MATLAB ode45 solver. Both methods have in common that the difference between the fourth- and fifth-order accurate solutions is calculated to determine the error, and to adapt the step size. The error estimate, e +, for the step is... 

Note that the Matlab ODE solver is given three arguments the function evaluating the right-hand side of the differential eqnation, the time interval to integrate over, and the initial conditions. The first line of code (opts) sets the relative error tolerance to le-4 (default is le-3). This line is followed by a command which calls the ode45 solver, and specifies that the differential equation is specified in the m-file. Fmther, the command also specifies that the ODE should be solved in the interval 0-1 and at the initial condition yo = 1 - The third line is just used to plot the solution in a graph. 

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