Let’s solve the linear ODE `u'=1.01u`

. First setup the package:

`<- diffeqr::diffeq_setup() de `

Define the derivative function `f(u,p,t)`

.

```
<- function(u,p,t) {
f return(1.01*u)
}
```

Then we give it an initial condition and a time span to solve over:

```
<- 1/2
u0 <- c(0., 1.) tspan
```

With those pieces we define the `ODEProblem`

and `solve`

the ODE:

```
= de$ODEProblem(f, u0, tspan)
prob = de$solve(prob) sol
```

This gives back a solution object for which `sol$t`

are the time points and `sol$u`

are the values. We can treat the solution as a continuous object in time via

and a high order interpolation will compute the value at `t=0.2`

. We can check the solution by plotting it:

`plot(sol$t,sol$u,"l")`

Now let’s solve the Lorenz equations. In this case, our initial condition is a vector and our derivative functions takes in the vector to return a vector (note: arbitrary dimensional arrays are allowed). We would define this as:

```
<- function(u,p,t) {
f = p[1]*(u[2]-u[1])
du1 = u[1]*(p[2]-u[3]) - u[2]
du2 = u[1]*u[2] - p[3]*u[3]
du3 return(c(du1,du2,du3))
}
```

Here we utilized the parameter array `p`

. Thus we use `diffeqr::ode.solve`

like before, but also pass in parameters this time:

```
<- c(1.0,0.0,0.0)
u0 <- list(0.0,100.0)
tspan <- c(10.0,28.0,8/3)
p <- de$ODEProblem(f, u0, tspan, p)
prob <- de$solve(prob) sol
```

The returned solution is like before except now `sol$u`

is an array of arrays, where `sol$u[i]`

is the full system at time `sol$t[i]`

. It can be convenient to turn this into an R matrix through `sapply`

:

`<- sapply(sol$u,identity) mat `

This has each row as a time series. `t(mat)`

makes each column a time series. It is sometimes convenient to turn the output into a `data.frame`

which is done via:

`<- as.data.frame(t(mat)) udf `

Now we can use `matplot`

to plot the timeseries together:

`matplot(sol$t,udf,"l",col=1:3)`

Now we can use the Plotly package to draw a phase plot:

`::plot_ly(udf, x = ~V1, y = ~V2, z = ~V3, type = 'scatter3d', mode = 'lines') plotly`