odeToVectorField

Reduce order of differential equations to first-order

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Support for character vector or string inputs will be removed in a future release. Instead, usesymsto declare variables, and replace inputs such asodeToVectorField('D2y = x')withsyms 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 equationseqn1,...,eqnNto a system of first-order differential equations, returned as a symbolic vector.

example

[V,S] = odeToVectorField(eqn1,...,eqnN)convertseqn1,...,eqnNand returns two symbolic vectors. The first vectorVis the same as the output of the previous syntax. The second vectorSshows the substitutions made to obtainV.

Examples

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二阶微分方程转换为冷杉t-Order System

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Define a second-order differential equation:

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

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

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

V =
                       
                         $(\begin{array}{c} Y_{2} \\ 3 t - t {Y_{1}}^{2} \end{array})$

The elements of V represent the system of first-order differential equations, whereV[i]= ${Y_{i}}^{'}$ and $Y_{1} = y$ . Here, the outputVrepresents these equations:

$\frac{d Y_{1}}{dt} = Y_{2}$

$\frac{{dY}_{2}}{dt} = 3 t - t Y_{1}^{2} .$

For details on the relation between the input and output, seeAlgorithms.

Return Substitutions Made When Reducing Order of Differential Equations

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

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

V =
                       
                         $(\begin{array}{c} Y_{2} \\ Y_{1} + Y_{3} \\ Y_{3} - Y_{1} \end{array})$

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

The elements ofVrepresent the system of first-order differential equations, whereV[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

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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 theode45function.

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

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

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

V =
                       
                         $(\begin{array}{c} Y_{2} \\ - ({Y_{1}}^{2} - 1) Y_{2} - Y_{1} \end{array})$

Generate a MATLAB function handle fromVby usingmatlabFunction.

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 usingode45.

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 oftby usinglinspace. Evaluate the solution for $y (t)$ , which is the first index inySol, by calling thedevalfunction with an index of 1. Plot the solution usingplot.

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

Figure contains an axes object. The axes object contains an object of type line.

Convert Second-Order Differential Equation with Initial Condition

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

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

V =
                       
                         $(\begin{array}{c} Y_{2} \\ x \end{array})$

Input Arguments

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`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 thedifffunction to indicate differentiation. For example, representd²y(t)/dt²=ty(t)by entering the following command.

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

Output Arguments

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`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 equationY[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 thatS(1) = Y[1],S(2) = Y[2],….

Tips

To solve the resulting system of first-order differential equations, generate a MATLAB^®function handle usingmatlabFunctionwithVas an input. Then, use the generated MATLAB function handle as an input for the MATLAB numerical solverode23orode45.
odeToVectorFieldcan convert only quasi-linear differential equations. That is, the highest-order derivatives must appear linearly. For example,odeToVectorFieldcan converty*y″(t) = –t²because it can be rewritten asy″(t) = –t²/y. However, it cannot converty″(t)²= –t²orsin(y″(t)) = –t².

Algorithms

To convert annth-order differential equation

$a_{n} (t) y^{(n)} + a_{n - 1} (t) y^{(n - 1)} + \dots + a_{1} (t) y^{'} + a_{0} (t) y + r (t) = 0$

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

$\begin{array}{l} Y_{1} = y \\ Y_{2} = y^{'} \\ Y_{3} = y^{″} \\ \dots \\ Y_{n - 1} = y^{(n - 2)} \\ Y_{n} = y^{(n - 1)} \end{array}$

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

$\begin{array}{l} Y_{1}^{'} = y^{'} = Y_{2} \\ Y_{2}^{'} = y^{″} = Y_{3} \\ \dots \\ Y_{n - 1}^{'} = y^{(n - 1)} = Y_{n} \\ Y_{n}^{'} = - \frac{a_{n - 1} (t)}{a_{n} (t)} Y_{n} - \frac{a_{n - 2} (t)}{a_{n} (t)} Y_{n - 1} - ... - \frac{a_{1} (t)}{a_{n} (t)} Y_{2} - \frac{a_{0} (t)}{a_{n} (t)} Y_{1} + \frac{r (t)}{a_{n} (t)} \end{array}$

odeToVectorFieldreturns the right sides of these equations as the elements of vectorVand the substitutions made as the second outputS.

Version History

Introduced in R2012a

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

Warns starting in R2019b

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

odeToVectorField

Syntax

Description

Examples

二阶微分方程转换为冷杉t-Order System

Return Substitutions Made When Reducing Order of Differential Equations

Solve Higher-Order Differential Equation Numerically

Convert Second-Order Differential Equation with Initial Condition

Input Arguments

eqn1,...,eqnN—Higher-order differential equationssymbolic differential equation|array of symbolic differential equations

Output Arguments

V— First-order differential equationssymbolic expression | vector of symbolic expressions

S— Substitutions in first-order equationsvector of symbolic expressions

Tips

Algorithms

Version History

R2019b:odeToVectorFieldwill no longer support character vector or string inputs

See Also

`eqn1,...,eqnN`—Higher-order differential equations
symbolic differential equation|array of symbolic differential equations

`V`— First-order differential equations
symbolic expression | vector of symbolic expressions

`S`— Substitutions in first-order equations
vector of symbolic expressions

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