If you follow these steps, then the ODE solver can solve the system of equations using a vector for the solution components, while the ODE function reshapes the vector into a matrix and solves each solution component for all of the initial conditions. įinally, the ODE function must reshape its output back into a vector so that the ODE solver receives a vector back from each function call. The ODE function must solve the equations in a vectorized format, so that the expression accepts vectors for the solution components. Each row of the matrix then contains all of the initial conditions for each variable. The ODE function must reshape the vector into a matrix with size s-by-n. Inside the ODE function, the solver passes the solution components p as a column vector. The ODE function must accept an extra input parameter for n, the number of initial conditions. Each column in the matrix then represents one complete set of initial conditions for the system. The size of the matrix is s-by- n, where s is the number of solution components and n is the number of initial conditions being solved for. Provide all of the initial conditions to ode45 as a matrix. Plot a phase plot with the results from all iterations. Create a vector of population sizes for y0, and then loop over the values to solve the equations for each set of initial conditions. This technique uses the same ODE function as the single initial condition technique, but the for-loop automates the solution process.įor example, you can hold the initial population size for x constant at 50, and use the for-loop to vary the initial population size for y between 10 and 400. The simplest way to solve a system of ODEs for multiple initial conditions is with a for-loop. Method 1: Compute Multiple Initial Conditions with for-loop The next two sections describe techniques to solve for many different initial conditions. However, this method only solves the equations for one initial condition at a time. To solve the equations for different initial population sizes, change the values in p0 and rerun the simulation. The resulting plots show the solution for the given initial population sizes.

System of odes