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# odeToVectorField

Reduce order of differential equations to 1

Character vector inputs will be removed in a future release. Instead, use `syms` to declare variables, and replace inputs such as ```odeToVectorField('D2y = x')``` with ```syms y(x), odeToVectorField(diff(y,x,2) == x)```.

## Syntax

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

## Description

example

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

example

``````[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 Higher-Order Equation to First-Order System

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

`$\frac{{d}^{2}y}{d{t}^{2}}+{y}^{2}t=3t.$`
```syms y(t) eqn = diff(y,2) + y^2*t == 3*t; V = odeToVectorField(eqn)```
```V = Y[2] 3*t - t*Y[1]^2```

The elements of `V` represent the system of differential equations because y = Y[1] and Y[i]′ = V[i]. Here, this particular output represents these equations:

• `diff(Y[1],t) = Y[2]`

• `diff(Y[2],t) = 3*t - t*Y[1]^2`

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

### Return Substitutions Made When Reducing Order

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 = Y[2] Y[1] + Y[3] Y[3] - Y[1] S = f Df g```

From `S`, we have `S[1] = Y[1] = f`, ```S[2] = Y[2] = diff(f)```, and `S[3] = Y[3] = g`.

### Numerically Solve Higher-Order Differential Equation

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

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

`$\frac{d{y}^{2}}{d{x}^{2}}=\left(1-{y}^{2}\right)\frac{dy}{dx}-y.$`
```syms y(t) eqn = diff(y,2) == (1-y^2)*diff(y)-y; V = odeToVectorField(eqn)```
```V = Y[2] - (Y[1]^2 - 1)*Y[2] - Y[1]```

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)]```

Solve this system over the interval `[0 20]` with initial conditions y’(0) = 2 and y’’(0) = 0 by using the `ode45` function.

```interval = [0 20]; y0 = [2 0]; ySol = ode45(M,interval,y0);```

Generate values of `t` in the interval by using the `linspace` function. For these values, evaluate the solution for `y`, which is the first index in `ySol`, by calling the `deval` function with an index of `1`. Plot the solution.

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

### Convert Higher-Order System with Initial Condition

Convert the second-order differential equation y″(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 = Y[2] x```

## Input Arguments

collapse all

Higher-order differential equations, specified as a symbolic differential equation or an array or comma-separated list 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)```.

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

## Output Arguments

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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].

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

`${a}_{n}\left(t\right){y}^{\left(n\right)}+{a}_{n-1}\left(t\right){y}^{\left(n-1\right)}+\dots +{a}_{1}\left(t\right){y}^{\prime }+{a}_{0}\left(t\right)y+r\left(t\right)=0$`

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

`$\begin{array}{l}{Y}_{1}=y\\ {Y}_{2}={y}^{\prime }\\ {Y}_{3}={y}^{″}\\ \dots \\ {Y}_{n-1}={y}^{\left(n-2\right)}\\ {Y}_{n}={y}^{\left(n-1\right)}\end{array}$`

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

`$\begin{array}{l}{Y}_{1}{}^{\prime }={y}^{\prime }={Y}_{2}\\ {Y}_{2}{}^{\prime }={y}^{″}={Y}_{3}\\ \dots \\ {Y}_{n-1}{}^{\prime }={y}^{\left(n-1\right)}={Y}_{n}\\ {Y}_{n}{}^{\prime }=-\frac{{a}_{n-1}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{n}-\frac{{a}_{n-2}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{n-1}-...-\frac{{a}_{1}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{2}-\frac{{a}_{0}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{1}+\frac{r\left(t\right)}{{a}_{n}\left(t\right)}\end{array}$`

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