Use `dsolve`

to compute
symbolic solutions to ordinary differential equations. Specify differential
equations as symbolic expressions containing `diff`

.

Before using `dsolve`

, create the symbolic
function for which you want to solve an ordinary differential equation.
Use `sym`

or `syms`

to create a
symbolic function. For example, create a function `y(x)`

:

syms y(x)

For details, see Create Symbolic Functions.

To specify initial or boundary conditions, use additional equations.
If you do not specify initial or boundary conditions, the solutions
will contain integration constants, such as `C1`

, `C2`

,
and so on.

For output from `dsolve`

, you can:

Call

`dsolve`

with the number of output variables equal to the number of dependent variables.Place the output in a structure whose fields contain the solutions of the differential equations.

Suppose you want to solve the equation `y'(t) = t*y`

.
First, create the symbolic function `y(t)`

:

syms y(t)

Now use `dsolve`

to solve the equation:

y(t) = dsolve(diff(y,t) == t*y)

y(t) = C2*exp(t^2/2)

`y(t) = C2*exp(t^2/2)`

is
a solution to the equation for any constant `C2`

.

Solve the same ordinary differential equation, but now specify
the initial condition `y(0) = 2`

:

syms y(t) y(t) = dsolve(diff(y,t) == t*y, y(0) == 2)

y(t) = 2*exp(t^2/2)

Nonlinear equations can have multiple solutions, even if you specify initial conditions. For example, solve this equation:

syms x(t) x(t) = dsolve((diff(x,t) + x)^2 == 1, x(0) == 0)

results in

x(t) = exp(-t) - 1 1 - exp(-t)

Solve this second-order differential equation with two initial
conditions. One initial condition is a derivative `y'(x)`

at ```
x
= 0
```

. To be able to specify this initial condition, create
an additional symbolic function `Dy = diff(y)`

. (You
also can use any valid function name instead of `Dy`

.)
Then `Dy(0) = 0`

specifies that `Dy = 0`

at ```
x
= 0
```

.

syms y(x) Dy = diff(y); y(x) = dsolve(diff(y, x, x) == cos(2*x) - y, y(0) == 1, Dy(0) == 0); y(x) = simplify(y)

y(x) = 1 - (8*sin(x/2)^4)/3

Solve this third-order ordinary differential equation:

$$\frac{{d}^{3}u}{d{x}^{3}}=u$$

$$u(0)=1,\text{\hspace{0.17em}}\text{}{u}^{\prime}(0)=-1,\text{}\text{\hspace{0.17em}}{{u}^{\prime}}^{\prime}(0)=\pi ,$$

Because the initial conditions contain the first- and the second-order
derivatives, create two additional symbolic functions, `Dy`

and `D2y`

to
specify these initial conditions:

syms u(x) Du = diff(u, x); D2u = diff(u, x, 2); u(x) = dsolve(diff(u, x, 3) == u, u(0) == 1, Du(0) == -1, D2u(0) == pi)

u(x) = (pi*exp(x))/3 - exp(-x/2)*cos((3^(1/2)*x)/2)*(pi/3 - 1) -... (3^(1/2)*exp(-x/2)*sin((3^(1/2)*x)/2)*(pi + 1))/3

This table shows examples of differential equations and their Symbolic Math Toolbox™ syntax. The last example is the Airy differential equation, whose solution is called the Airy function.

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