diff --git a/Project.toml b/Project.toml
index c525894c..7e8f4f0a 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "ImplicitDifferentiation"
 uuid = "57b37032-215b-411a-8a7c-41a003a55207"
 authors = ["Guillaume Dalle", "Mohamed Tarek"]
-version = "0.9.0"
+version = "0.9.1"
 
 [deps]
 ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
diff --git a/ext/ImplicitDifferentiationChainRulesCoreExt.jl b/ext/ImplicitDifferentiationChainRulesCoreExt.jl
index c86bafa4..4b3d0397 100644
--- a/ext/ImplicitDifferentiationChainRulesCoreExt.jl
+++ b/ext/ImplicitDifferentiationChainRulesCoreExt.jl
@@ -14,17 +14,18 @@ using ImplicitDifferentiation:
 # not covered by Codecov for now
 ImplicitDifferentiation.chainrules_suggested_backend(rc::RuleConfig) = AutoChainRules(rc)
 
-struct ImplicitPullback{TA,TB,TL,TP,Nargs}
+struct ImplicitPullback{TA,TB,TL,TC,TP,Nargs}
     Aᵀ::TA
     Bᵀ::TB
     linear_solver::TL
+    c0::TC
     project_x::TP
     _Nargs::Val{Nargs}
 end
 
-function (pb::ImplicitPullback{TA,TB,TL,TP,Nargs})((dy, dz)) where {TA,TB,TL,TP,Nargs}
-    (; Aᵀ, Bᵀ, linear_solver, project_x) = pb
-    dc = linear_solver(Aᵀ, -unthunk(dy))
+function (pb::ImplicitPullback{TA,TB,TL,TC,TP,Nargs})((dy, dz)) where {TA,TB,TL,TP,TC,Nargs}
+    (; Aᵀ, Bᵀ, linear_solver, c0, project_x) = pb
+    dc = linear_solver(Aᵀ, -unthunk(dy), c0)
     dx = Bᵀ(dc)
     df = NoTangent()
     dargs = ntuple(unimplemented_tangent, Val(Nargs))
@@ -37,6 +38,7 @@ function ChainRulesCore.rrule(
     (; conditions, linear_solver) = implicit
     y, z = implicit(x, args...)
     c = conditions(x, y, z, args...)
+    c0 = zero(c)
 
     suggested_backend = chainrules_suggested_backend(rc)
     prep = ImplicitFunctionPreparation(eltype(x))
@@ -44,7 +46,7 @@ function ChainRulesCore.rrule(
     Bᵀ = build_Bᵀ(implicit, prep, x, y, z, c, args...; suggested_backend)
     project_x = ProjectTo(x)
 
-    implicit_pullback = ImplicitPullback(Aᵀ, Bᵀ, linear_solver, project_x, Val(N))
+    implicit_pullback = ImplicitPullback(Aᵀ, Bᵀ, linear_solver, c0, project_x, Val(N))
     return (y, z), implicit_pullback
 end
 
diff --git a/ext/ImplicitDifferentiationForwardDiffExt.jl b/ext/ImplicitDifferentiationForwardDiffExt.jl
index 5228c270..54fa761f 100644
--- a/ext/ImplicitDifferentiationForwardDiffExt.jl
+++ b/ext/ImplicitDifferentiationForwardDiffExt.jl
@@ -11,6 +11,7 @@ function (implicit::ImplicitFunction)(
     x = value.(x_and_dx)
     y, z = implicit(x, args...)
     c = implicit.conditions(x, y, z, args...)
+    y0 = zero(y)
 
     suggested_backend = AutoForwardDiff()
     A = build_A(implicit, prep, x, y, z, c, args...; suggested_backend)
@@ -21,7 +22,7 @@ function (implicit::ImplicitFunction)(
     end
     dC = map(B, dX)
     dY = map(dC) do dₖc
-        dₖy = implicit.linear_solver(A, -dₖc)
+        dₖy = implicit.linear_solver(A, -dₖc, y0)
         return dₖy
     end
 
diff --git a/src/settings.jl b/src/settings.jl
index c54e2a33..a917e21b 100644
--- a/src/settings.jl
+++ b/src/settings.jl
@@ -12,7 +12,7 @@ Specify that linear systems `Ax = b` should be solved with a direct method.
 """
 struct DirectLinearSolver end
 
-function (solver::DirectLinearSolver)(A, b::AbstractVector)
+function (solver::DirectLinearSolver)(A, b::AbstractVector, x0::AbstractVector)
     return A \ b
 end
 
@@ -33,8 +33,8 @@ struct IterativeLinearSolver{K}
     end
 end
 
-function (solver::IterativeLinearSolver)(A, b)
-    sol, info = linsolve(A, b; solver.kwargs...)
+function (solver::IterativeLinearSolver)(A, b, x0)
+    sol, info = linsolve(A, b, x0; solver.kwargs...)
     @assert info.converged == 1
     return sol
 end