Symbolic Math
Toolbox™ offers both symbolic and numeric equation
solvers. This topic shows you how to solve an equation symbolically
using the symbolic solver `solve`

. To compare symbolic
and numeric solvers, see Select Numeric or Symbolic Solver.

If `eqn`

is an equation, ```
solve(eqn,
x)
```

solves `eqn`

for the symbolic variable `x`

.

Use the `==`

operator to specify the familiar
quadratic equation and solve it using `solve`

.

syms a b c x eqn = a*x^2 + b*x + c == 0; solx = solve(eqn, x)

solx = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a)

`solx`

is a symbolic vector containing the
two solutions of the quadratic equation. If the input `eqn`

is
an expression and not an equation, `solve`

solves
the equation `eqn == 0`

.

To solve for a variable other than `x`

, specify
that variable instead. For example, solve `eqn`

for `b`

.

solb = solve(eqn, b)

solb = -(a*x^2 + c)/x

If you do not specify a variable, `solve`

uses `symvar`

to
select the variable to solve for. For example, `solve(eqn)`

solves `eqn`

for `x`

.

`solve`

does not automatically return all
solutions of an equation. Solve the equation `cos(x) == -sin(x)`

.
The `solve`

function returns one of many solutions.

syms x solx = solve(cos(x) == -sin(x), x)

solx = -pi/4

To return all solutions along with the parameters in the solution
and the conditions on the solution, set the `ReturnConditions`

option
to `true`

. Solve the same equation for the full
solution. Provide three output variables: for the solution to `x`

,
for the parameters in the solution, and for the conditions on the
solution.

syms x [solx, param, cond] = solve(cos(x) == -sin(x), x, 'ReturnConditions', true)

solx = pi*k - pi/4 param = k cond = in(k, 'integer')

`solx`

contains the solution for `x`

,
which is `pi*k - pi/4`

. The `param`

variable
specifies the parameter in the solution, which is `k`

.
The `cond`

variable specifies the condition ```
in(k,
'integer')
```

on the solution, which means `k`

must
be an integer. Thus, `solve`

returns a periodic
solution starting at `pi/4`

which repeats at intervals
of `pi*k`

, where `k`

is an integer.

You can use the solutions, parameters, and conditions returned
by `solve`

to find solutions within an interval
or under additional conditions.

To find values of `x`

in the interval `-2*pi<x<2*pi`

,
solve `solx`

for `k`

within that
interval under the condition `cond`

. Assume the condition `cond`

using `assume`

.

assume(cond) solk = solve(-2*pi<solx, solx<2*pi, param)

solk = -1 0 1 2

To find values of `x`

corresponding to these
values of `k`

, use `subs`

to substitute
for `k`

in `solx`

.

xvalues = subs(solx, solk)

xvalues = -(5*pi)/4 -pi/4 (3*pi)/4 (7*pi)/4

To convert these symbolic values into numeric values for use
in numeric calculations, use `vpa`

.

xvalues = vpa(xvalues)

xvalues = -3.9269908169872415480783042290994 -0.78539816339744830961566084581988 2.3561944901923449288469825374596 5.4977871437821381673096259207391

The previous sections used `solve`

to solve
the equation `cos(x) == -sin(x)`

. The solution to
this equation can be visualized using plotting functions such as `fplot`

and `scatter`

.

Plot both sides of equation `cos(x) == -sin(x)`

.

fplot(cos(x)) hold on grid on fplot(-sin(x)) title('Both sides of equation cos(x) = -sin(x)') legend('cos(x)','-sin(x)','Location','best','AutoUpdate','off')

Calculate the values of the functions at the values of `x`

,
and superimpose the solutions as points using `scatter`

.

yvalues = cos(xvalues)

yvalues =$$\left(\begin{array}{c}-0.70710678118654752440084436210485\\ 0.70710678118654752440084436210485\\ -0.70710678118654752440084436210485\\ 0.70710678118654752440084436210485\end{array}\right)$$

scatter(xvalues, yvalues)

As expected, the solutions appear at the intersection of the two plots.

If results look complicated, `solve`

is stuck,
or if you want to improve performance, see, Troubleshoot Equation Solutions from solve Function.