Next Step Suggestions for Symbolic Workflows in Live Editor

Simplify Symbolic Expression

Create a symbolic expression that contains exponential functions and imaginary numbers. Run the following code to generate the symbolic output.

syms x
expr = 1i*(exp(-1i*x) - exp(1i*x))/(exp(-1i*x) + exp(1i*x))

expr = 
$$\frac{{\mathrm{e}}^{-x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}-{\mathrm{e}}^{x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{{\mathrm{e}}^{-x\hspace{0.17em}\mathrm{i}}+{\mathrm{e}}^{x\hspace{0.17em}\mathrm{i}}}$$

To simplify the expression, right-click the symbolic output and select Rewriting and simplifying expressions > Simplify expression.

Live Editor inserts and applies the Simplify Symbolic Expression task to interactively simplify or rearrange symbolic expressions. Change the computational effort to Medium to get a simpler result.

Substitute Coefficients and Solve Quadratic Equation

Create a quadratic equation with coefficients $$a$$, $$b$$, and $$c$$. Run the following code to generate the symbolic output.

syms a b c x
eqn = a*x^2 + b*x + c == 0

eqn = $$a\hspace{0.17em}{x}^2+b\hspace{0.17em}x+c=0$$

To substitute the coefficients in the equation, right-click the output and select Substitute variables.

Live Editor inserts the subs function to substitute the coefficients and variables in the equations. For the subs function, Live Editor does not run the function automatically. To substitute $$a=3$$, $$b=2$$, and $$c=0$$, change the second and third arguments of the subs function to [a,b,c] and [3,2,0]. Run the Live Editor section afterward to apply the subs function.

var3 = subs(eqn,[a,b,c],[3,2,0])

var3 = $$3\hspace{0.17em}{x}^2+2\hspace{0.17em}x=0$$

To solve the quadratic equation, right-click the output and select Solve equation analytically.

Live Editor inserts and applies the Solve Symbolic Equation task to interactively find analytic solutions of symbolic equations.

Plot Explicit and Implicit Functions

Create three symbolic variables x, y, and z and a sinusoidal function. Run the following code to generate the symbolic output.

syms x y z
f = sin(2*x)

f = $$\sin \left(2\hspace{0.17em}x\right)$$

To plot the sinusoidal function, right-click the output and select Plotting functions > Plot function.

Live Editor plots the sinusoidal function using fplot.

fplot(f)

Next, create an equation that represents a hyperbola. Run the following code to generate the symbolic output.

eqn = x^2 - y^2 == 1

eqn = $$x^2-y^2=1$$

To plot the sinusoidal function, right-click the output and select Plotting functions > Plot implicit equation in 2-D.

Live Editor plots the hyperbola using fimplicit.

fimplicit(eqn)

Next, create a symbolic function that represents a torus. Run the following code to generate the symbolic output.

f(x,y,z) = (x^2 + y^2 + z^2 + 3^2 - 2^2)^2 - 4*3^2*(x^2+y^2)

f(x, y, z) = $${\left(x^2+y^2+z^2+5\right)}^2-36\hspace{0.17em}{x}^2-36\hspace{0.17em}{y}^2$$

To plot the torus, right-click the output and select Plot implicit equation in 3-D.

Live Editor plots the torus using fimplicit3.

fimplicit3(f)

Apply Matrix Functions

Create three symbolic variables x, y, and z and a symbolic matrix. Run the following code to generate the symbolic output.

syms x y z
M1 = sym([x^2 + a, x; y + 2, 3*y^2])

M1 = 
$$\left(\begin{array}{cc}{x}^2+a& x\\ y+2& 3\hspace{0.17em}{y}^2\end{array}\right)$$

To invert the matrix, right-click the output and select Applying matrix functions > Invert matrix. Note that the Applying matrix functions > Invert matrix suggestion is available only if the symbolic output is a symbolic matrix.

Live Editor inserts and applies the inv function to invert the matrix.

var4 = inv(M1)

var4 = 
$$\begin{array}{l}\left(\begin{array}{cc}-\frac{3\hspace{0.17em}{y}^2}{{\sigma }_1}& \frac{x}{{\sigma }_1}\\ \frac{y+2}{{\sigma }_1}& -\frac{x^2+a}{{\sigma }_1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=-3\hspace{0.17em}{x}^2\hspace{0.17em}{y}^2+x\hspace{0.17em}y+2\hspace{0.17em}x-3\hspace{0.17em}a\hspace{0.17em}{y}^2\end{array}$$

Next, create a 1-by-3 symbolic vector. Run the following code to generate the symbolic output.

M2 = sym([x*y*z, y^2, x + z])

M2 = $$\left(\begin{array}{ccc}x\hspace{0.17em}y\hspace{0.17em}z& y^2& x+z\end{array}\right)$$

To compute the Jacobian of the vector, right-click the output and select Applying matrix functions > Compute Jacobian matrix of vector. Note that the Applying matrix functions > Compute Jacobian matrix of vector suggestion is available only if the symbolic output is a symbolic vector.

Live Editor inserts and applies the jacobian function to compute the Jacobian of the vector.

var5 = jacobian(M2)

var5 = 
$$\left(\begin{array}{ccc}y\hspace{0.17em}z& x\hspace{0.17em}z& x\hspace{0.17em}y\\ 0& 2\hspace{0.17em}y& 0\\ 1& 0& 1\end{array}\right)$$

Next, create another 1-by-3 symbolic vector. Run the following code to generate the symbolic output.

M3 = sym([x^2*y, 2*x, z])

M3 = $$\left(\begin{array}{ccc}{x}^2\hspace{0.17em}y& 2\hspace{0.17em}x& z\end{array}\right)$$

To compute the curl of the vector, right-click the output and select Applying matrix functions > Compute curl of vector field.

Live Editor inserts and applies the curl function to compute the curl of the vector.

var6 = curl(M3)

var6 = 
$$\left(\begin{array}{c}0\\ 0\\ 2-x^2\end{array}\right)$$

Solve Differential Equation, Compute Integral Transform, and Find Poles

Create a second-order differential equation. Run the following code to generate the symbolic output.

syms f(x) s
eqn = diff(f,x,x) == -9*f

eqn(x) = 
$$\frac{{\partial }^2}{\partial x^2}\mathrm{ }f\left(x\right)=-9\hspace{0.17em}f\left(x\right)$$

To solve the differential equation, right-click the output and select Solve differential equation. Note that the Solve differential equation suggestion is available only if the symbolic output is a differential equation.

Live Editor inserts and applies the dsolve function to solve the differential equation.

var7 = dsolve(eqn)

var7 = $$C_1\hspace{0.17em}\cos \left(3\hspace{0.17em}x\right)-C_2\hspace{0.17em}\sin \left(3\hspace{0.17em}x\right)$$

Next, find the Laplace transform of the solution. Right-click the output of dsolve and select Computing integral transforms > Compute Laplace transform.

Live Editor inserts and applies the laplace function to compute the Laplace transform.

var8 = laplace(var7)

var8 = 
$$\frac{C_1\hspace{0.17em}s}{s^2+9}-\frac{3\hspace{0.17em}{C}_2}{s^2+9}$$

Finally, find the poles of the Laplace transform. Right-click the output of the Laplace transform and select Applying calculus functions > Compute poles of function.

Live Editor inserts and applies the poles function to compute the poles.

var9 = poles(var8,s)

var9 = 
$$\left(\begin{array}{c}-3\hspace{0.17em}\mathrm{i}\\ 3\hspace{0.17em}\mathrm{i}\end{array}\right)$$

Convert Units and Check Consistency of Units

Create a ratio of two lengths with different units. Run the following code to generate the symbolic output.

u = symunit;
ratio = (7*u.mi)/(420*u.ft)

ratio = 
$$\frac{1}{60}\hspace{0.17em}\frac{\mathrm{mi}\mathrm{"mile\; -\; a\; physical\; unit\; of\; length."}}{\mathrm{ft}\mathrm{"foot\; -\; a\; physical\; unit\; of\; length."}}$$

Convert the units to the meter-gram-second system. Right-click the output and select Applying physical unit functions > Convert units. Note that the Applying physical unit functions suggestion is available only if the symbolic output contains symbolic units.

Live Editor inserts and applies the unitConvert function to convert the units to the meter-gram-second system.

var10 = unitConvert(ratio,[symunit('m'),symunit('g'),symunit('s')])

var10 = $$88$$

Next, create two symbolic expressions x and y that describe the $$x$$- and $$y$$-coordinates of a moving projectile. Create a symbolic equation r that compares the units of x and y to the length units m and ft. Run the following code to generate the symbolic output.

syms theta ts x y r
g = 9.81*u.m/u.s^2;
v = 10*u.m/u.s;
t = ts*u.s;
x = v*cos(theta)*t;
y = v*sin(theta)*t + (-g*t^2)/2;
r = [x == u.m y == u.ft]

r = 
$$\left(\begin{array}{cc}10\hspace{0.17em}\mathrm{ts}\hspace{0.17em}\cos \left(\theta \right)\hspace{0.17em}\mathrm{m}\mathrm{"meter\; -\; a\; physical\; unit\; of\; length."}=\mathrm{m}\mathrm{"meter\; -\; a\; physical\; unit\; of\; length."}& 10\hspace{0.17em}\mathrm{ts}\hspace{0.17em}\sin \left(\theta \right)\hspace{0.17em}\mathrm{m}\mathrm{"meter\; -\; a\; physical\; unit\; of\; length."}-\frac{981\hspace{0.17em}{\mathrm{ts}}^2}{200}\hspace{0.17em}\mathrm{m}\mathrm{"meter\; -\; a\; physical\; unit\; of\; length."}=\mathrm{ft}\mathrm{"foot\; -\; a\; physical\; unit\; of\; length."}\end{array}\right)$$

Check the consistency of units in r. Right-click the output and select Applying physical unit functions > Check consistency of units.

Live Editor inserts and applies the checkUnits function to check the consistency and compatibility of the units in r.

The checkUnits function returns a structure containing the fields Consistent and Compatible. The Consistent field returns logical 1(true) if all terms in r have the same dimension and same unit with a conversion factor of 1. The Compatible field returns logical 1(true) if all terms have the same dimension, but not necessarily the same unit.

var11 = checkUnits(r)

var11 = struct with fields:
    Consistent: [1 0]
    Compatible: [1 1]