odeToVectorField

Reduce order of differential equations to first-order

Syntax

V = odeToVectorField(eqn1,...,eqnN)

[V,S] = odeToVectorField(eqn1,...,eqnN)

Description

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:

$$\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dt}}^2}+{\mathit{y}}^2\mathit{t}=3\mathit{t}.$$

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 = 
$$\left(\begin{array}{c}{Y}_2\\ 3\hspace{0.17em}t-t\hspace{0.17em}{Y_1}^2\end{array}\right)$$

The elements of V represent the system of first-order differential equations, where V[i] = $${Y_i}^{\prime }$$ and $$Y_1=y$$. Here, the output V represents these equations:

$$\frac{\mathit{d}{\mathit{Y}}_1}{\mathit{dt}}={\mathit{Y}}_2$$

$$\frac{{\mathit{dY}}_2}{\mathit{dt}}=3\mathit{t}-\mathit{t}{\mathit{Y}}_1^2.$$

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] = $${Y_i}^{\prime }$$. 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.

$$\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{t}}^2}=\left(1-{\mathit{y}}^2\right)\frac{\mathit{dy}}{\mathit{dt}}-\mathit{y}.$$

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

V = 
$$\left(\begin{array}{c}{Y}_2\\ -\left({Y_1}^2-1\right)\hspace{0.17em}{Y}_2-Y_1\end{array}\right)$$

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 $$y^{\prime }(0)=2$$ and $$y^{\prime \prime }(0)=0$$. 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)

Convert Second-Order Differential Equation with Initial Condition

Convert the second-order differential equation $$y^{\prime \prime }(x)=x$$ with the initial condition $$y(0)=a$$ to a first-order system.

syms y(x) a
eqn = diff(y,x,2) == x;
cond = y(0) == a;
V = odeToVectorField(eqn,cond)

V = 
$$\left(\begin{array}{c}{Y}_2\\ x\end{array}\right)$$

Input Arguments

eqn1,...,eqnN — Higher-order differential equations

symbolic differential equation | array of symbolic differential equations

Higher-order differential equations, specified as a symbolic differential equation or an array of symbolic differential equations. Use the == operator to create an equation. Use the diff function to indicate differentiation. For example, represent d²y(t)/dt² = t y(t) by entering the following command.

syms y(t)
eqn = diff(y,2) == t*y;

Output Arguments

V — First-order differential equations

symbolic expression | vector of symbolic expressions

First-order differential equations, returned as a symbolic expression or a vector of symbolic expressions. Each element of this vector is the right side of the first-order differential equation Y[i]′ = V[i].

S — Substitutions in first-order equations

vector of symbolic expressions

Substitutions in first-order equations, returned as a vector of symbolic expressions. The elements of the vector represent the substitutions, such that S(1) = Y[1], S(2) = Y[2],….

Tips

• To solve the resulting system of first-order differential equations, generate a MATLAB^® function handle using matlabFunction with V as an input. Then, use the generated MATLAB function handle as an input for the MATLAB numerical solver ode23 or ode45.

• odeToVectorField can convert only quasi-linear differential equations. That is, the highest-order derivatives must appear linearly. For example, odeToVectorField can convert y*y″(t) = –t² because it can be rewritten as y″(t) = –t²/y. However, it cannot convert y″(t)² = –t² or sin(y″(t)) = –t².

Algorithms

To convert an nth-order differential equation

$$a_n(t)y^{(n)}+a_{n-1}(t)y^{(n-1)}+\dots +a_1(t)y^{\prime }+a_0(t)y+r(t)=0$$

into a system of first-order differential equations, odetovectorfield makes these substitutions.

$$\begin{array}{l}{Y}_1=y\\ Y_2=y^{\prime }\\ Y_3=y^{″}\\ \dots \\ Y_{n-1}=y^{(n-2)}\\ Y_n=y^{(n-1)}\end{array}$$

Using the new variables, it rewrites the equation as a system of n first-order differential equations:

$$\begin{array}{l}{Y}_1{}^{\prime }=y^{\prime }=Y_2\\ Y_2{}^{\prime }=y^{″}=Y_3\\ \dots \\ Y_{n-1}{}^{\prime }=y^{(n-1)}=Y_n\\ Y_n{}^{\prime }=-\frac{a_{n-1}\left(t\right)}{a_n\left(t\right)}{Y}_n-\frac{a_{n-2}\left(t\right)}{a_n\left(t\right)}{Y}_{n-1}-\mathrm{...}-\frac{a_1\left(t\right)}{a_n\left(t\right)}{Y}_2-\frac{a_0\left(t\right)}{a_n\left(t\right)}{Y}_1+\frac{r\left(t\right)}{a_n\left(t\right)}\end{array}$$

odeToVectorField returns the right sides of these equations as the elements of vector V and the substitutions made as the second output S.

Version History

Introduced in R2012a

R2019b: odeToVectorField will no longer support character vector or string inputs

odeToVectorField will not accept equations as strings or character vectors in a future release. Instead, use symbolic expressions or sym objects to define differential equations. For example, replace inputs such as odeToVectorField('D2y=x') with syms y(x), odeToVectorField(diff(y,x,2)==x).