 Solution

ID: past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2010/a/1/ii/c/solution

The distributional derivative is defined by moving the derivative onto a test function and changing the sign:

⟨ S^{'}, ϕ ⟩ = - ⟨ S, ϕ^{'} ⟩ .

Since

⟨ δ_{a}, ϕ ⟩ = ϕ (a)

, one has

⟨ δ_{0}^{'}, ϕ ⟩ = - ϕ^{'} (0)

. Hence

T = - δ_{0}^{'} + δ_{1} = - δ^{'} (x) + δ (x - 1) .

The minus sign in front of

δ_{0}^{'}

is essential: it makes its action the positive derivative evaluation required here.

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