Remove deepcopy in solve! for DualLinearCache, put Dual types in type system

[jClugstor]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Trying to address SciML/OrdinaryDiffEq.jl#2837 and #648

This gets rid of the call to deepcopy.

This also puts the type of the dual number that needs to be constructed in to the type system instead of having it be a field in the dual caches. I'm not sure if that will help with trimming but I think in general it's a better way to do this.

[topolarity]

Looks like there are still some deepcopys in __init:

Stacktrace:
  [1] deepcopy_internal(x::Any, stackdict::IdDict{Any, Any})
    @ Base deepcopy.jl:69
  [2] deepcopy(x::LinearAlgebra.Symmetric{Float64, Matrix{Float64}})
    @ Base deepcopy.jl:29 [inlined]
  [3] __init(::SciMLBase.LinearProblem{Vector{Float64}, true, LinearAlgebra.Symmetric{Float64, Matrix{Float64}}, Vector{Float64}, SciMLBase.NullParameters, Nothing, @Kwargs{abstol::Float64, reltol::Float64}}, ::LinearSolve.CholeskyFactorization{LinearAlgebra.NoPivot, Nothing}; alias::SciMLBase.LinearAliasSpecifier, abstol::Float64, reltol::Float64, maxiters::Int64, verbose::Bool, Pl::Nothing, Pr::Nothing, assumptions::LinearSolve.OperatorAssumptions{Bool}, sensealg::LinearSolve.LinearSolveAdjoint{Missing}, kwargs::@Kwargs{})
    @ LinearSolve ~/repos/LinearSolve.jl/src/common.jl:324 [inlined]
...

[jClugstor]

I should have named this "Remove deepcopy in solve! for DualLinearCache" etc. . I wasn't looking at any of the other deepcopy calls.

[topolarity]

Makes sense, thanks for the clarification - that probably means #648 should be left open