Fix BDF allocation tests to use correct problem types

Properly separated BDF solvers by their required problem types:
- Standard BDF methods (ABDF2, QNDF, QBDF, FBDF, SBDF, MEBDF2): use ODEProblem
- IMEX methods (IMEXEuler, IMEXEulerARK): use SplitODEProblem
- DAE methods (DABDF2, DImplicitEuler, DFBDF): use DAEProblem

This matches the LowOrderRK test structure and ensures each solver
is tested with the appropriate problem type it was designed for.

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <[email protected]>

Author: ChrisRackauckas

lib/OrdinaryDiffEqBDF/test/allocation_tests.jl

@@ -10,7 +10,7 @@ Currently, many BDF solvers are allocating and marked with @test_broken.
 """
 
 @testset "BDF Allocation Tests" begin
-    # Test problem for BDF methods
+    # Test problem for standard ODE BDF methods
     function simple_system!(du, u, p, t)
         du[1] = -0.5 * u[1]
         du[2] = -1.5 * u[2]
@@ -28,14 +28,26 @@ Currently, many BDF solvers are allocating and marked with @test_broken.
     end
     split_prob = SplitODEProblem(f1!, f2!, [1.0, 1.0], (0.0, 1.0))
 
+    # DAE problem for DAE solvers
+    function dae_f!(du, u, p, t)
+        du[1] = -0.5 * u[1] + u[2]
+        du[2] = u[1] - u[2]
+    end
+    du0 = zeros(2)
+    differential_vars = [true, false]
+    dae_prob = DAEProblem(dae_f!, du0, [1.0, 1.0], (0.0, 1.0), differential_vars=differential_vars)
+    
     # Test all exported BDF solvers for allocation-free behavior
+    # Standard ODE BDF methods
     bdf_solvers = [ABDF2(), QNDF1(), QBDF1(), QNDF2(), QBDF2(), QNDF(), QBDF(), FBDF(),
-                   SBDF(order=2), SBDF2(), SBDF3(), SBDF4(), MEBDF2(), 
-                   DABDF2(), DImplicitEuler(), DFBDF()]
+                   SBDF(order=2), SBDF2(), SBDF3(), SBDF4(), MEBDF2()]
 
-    # IMEX methods need special handling with split problem
+    # IMEX methods need SplitODEProblem
     imex_solvers = [IMEXEuler(), IMEXEulerARK()]
 
+    # DAE methods need DAEProblem
+    dae_solvers = [DABDF2(), DImplicitEuler(), DFBDF()]
+    
     @testset "BDF Solver Allocation Analysis" begin
         for solver in bdf_solvers
             @testset "$(typeof(solver)) allocation check" begin
@@ -85,4 +97,29 @@ Currently, many BDF solvers are allocating and marked with @test_broken.
             end
         end
     end
+    
+    @testset "DAE Solver Allocation Analysis" begin
+        for solver in dae_solvers
+            @testset "$(typeof(solver)) allocation check" begin
+                integrator = init(dae_prob, solver, dt=0.1, save_everystep=false, abstol=1e-6, reltol=1e-6)
+                step!(integrator)  # Setup step may allocate
+                
+                # Use AllocCheck to verify step! is allocation-free
+                allocs = check_allocs(step!, (typeof(integrator),))
+                
+                # These solvers should be allocation-free, but mark as broken for now
+                # to verify with AllocCheck (more accurate than @allocated)
+                @test length(allocs) == 0 broken=true
+                
+                if length(allocs) > 0
+                    println("AllocCheck found $(length(allocs)) allocation sites in $(typeof(solver)) step!:")
+                    for (i, alloc) in enumerate(allocs[1:min(3, end)])  # Show first 3
+                        println("  $i. $alloc")
+                    end
+                else
+                    println("✓ $(typeof(solver)) appears allocation-free with AllocCheck")
+                end
+            end
+        end
+    end
 end