odeToVectorField

V = odeToVectorField( eqn1. eqnN ) converts higher-order differential equations eqn1. eqnN to a system of first-order differential equations, returned as a symbolic vector.

[ V , S ] = odeToVectorField( eqn1. eqnN ) converts eqn1. eqnN and returns two symbolic vectors. The first vector V is the same as the output of the previous syntax. The second vector S shows the substitutions made to obtain V .

Examples

Convert Second-Order Differential Equation to First-Order System

Define a second-order differential equation:

d 2 y dt 2 + y 2 t = 3 t .

Convert the second-order differential equation to a system of first-order differential equations.

syms y(t) eqn = diff(y,2) + y^2*t == 3*t; V = odeToVectorField(eqn)

V = 
$$\left(\begin{array}{c}{Y}_2\\ 3t-t{Y_1}^2\end{array}\right)$$

The elements of V represent the system of first-order differential equations, where V[i] = Y i ′ and Y 1 = y . Here, the output V represents these equations:

dY 2 dt = 3 t - t Y 1 2 .

For details on the relation between the input and output, see Algorithms.

Return Substitutions Made When Reducing Order of Differential Equations

When reducing the order of differential equations, return the substitutions that odeToVectorField makes by specifying a second output argument.

syms f(t) g(t) eqn1 = diff(g) == g-f; eqn2 = diff(f,2) == g+f; eqns = [eqn1 eqn2]; [V,S] = odeToVectorField(eqns)

V = 
$$\left(\begin{array}{c}{Y}_2\\ Y_1+Y_3\\ Y_3-Y_1\end{array}\right)$$

S = 
$$\left(\begin{array}{c}f\\ \mathrm{Df}\\ g\end{array}\right)$$

The elements of V represent the system of first-order differential equations, where V[i] = Y i ′ . The output S shows the substitutions being made, S[1] = Y 1 = f , S[2] = Y 2 = diff(f) , and S[3] = Y 3 = g .

Solve Higher-Order Differential Equation Numerically

Solve a higher-order differential equation numerically by reducing the order of the equation, generating a MATLAB® function handle, and then finding the numerical solution using the ode45 function.

Convert the following second-order differential equation to a system of first-order differential equations by using odeToVectorField .

d 2 y d t 2 = ( 1 - y 2 ) dy dt - y .

syms y(t) eqn = diff(y,2) == (1-y^2)*diff(y)-y; V = odeToVectorField(eqn)

V = 
$$\left(\begin{array}{c}{Y}_2\\ -\left({Y_1}^2-1\right)Y_2-Y_1\end{array}\right)$$

Generate a MATLAB function handle from V by using matlabFunction .

M = matlabFunction(V,'vars','t','Y'>)

M = function_handle with value: @(t,Y)[Y(2);-(Y(1).^2-1.0).*Y(2)-Y(1)]

Specify the solution interval to be [0 20] and the initial conditions to be y ′ ( 0 ) = 2 and y ′ ′ ( 0 ) = 0 . Solve the system of first-order differential equations by using ode45 .

interval = [0 20]; yInit = [2 0]; ySol = ode45(M,interval,yInit);

Next, plot the solution y ( t ) within the interval t = [0 20] . Generate the values of t by using linspace . Evaluate the solution for y ( t ) , which is the first index in ySol , by calling the deval function with an index of 1. Plot the solution using plot .

tValues = linspace(0,20,100); yValues = deval(ySol,tValues,1); plot(tValues,yValues)