### 4.3.5 Approximate solution of the system v’=f(t,v) : `odesolve`

- If v is a vector of variables [x1,..,xn] and if f is given by a vector of expressions `[e1,...,en]` depending of t and of [x1,..,xn], if the init value of v at `t0` is the vector [x10,...,xn0] then the instruction
  `odesolve([e1,..,en],t=t0..t1,[x1,...,xn], [x10,...,xn0])`
  returns an approximated value of v at t=t1. With the optional argument `curve`, `odesolve` returns the list of the intermediate values of [t,v(t)] computed by the solver.

  Example, to solve the system

  x′(t) | = | −y(t)
  y′(t) | = | x(t)

  input :

  `odesolve([-y,x],t=0..pi,[x,y],[0,1])`

  Output :

  `[-1.79045146764e-15,-1]`
- If f is a function from ℝ × ℝn to ℝn.
  `odesolve(t0..t1,(t,v)->f(t,v),v0)` or
  `odesolve(t0..t1,f,v0)`
  computes an approximate value of v(t1) where the vector v(t) in ℝn is the solution of

  v′(t)=f(t,v(t)), v(t0)=v0

  With the optional argument `curve`, `odesolve` returns the list of the intermediate value [t,v(t)] computed by the solver.

  Example, to solve the system :

  x′(t) | = | −y(t)
  y′(t) | = | x(t)

  Input :

  `odesolve(0..pi,(t,v)->[-v[1],v[0]],[0,1])`

  Or define the function:

  `f(t,v):=[-v[1],v[0]]`

  then input :

  `odesolve(0..pi,f,[0,1])`

  Output :

  `[-1.79045146764e-15,-1]`

  Alternative input :

  `odesolve(0..pi/4,f,[0,1],curve)`

  Output :

  `[[0.1781,[-0.177159948386,0.984182072936]], [0.3781,[-0.369155338156,0.929367707805]], [0.5781,[-0.54643366953,0.837502384954]], [0.7781,[-0.701927414872,0.712248484906]]]`