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I'd be very happy if you could help me solve a problem in my research.

I need to prove the following:

[itex] H'(r) = -y(r) - k H(r) [/itex]

k is a constant.

y is strictly increasing, but not continuous.

Let [itex] (a,b]\subset R [/itex].

[itex] (H_x, y_x) [/itex] denotes solution x.

[itex] H_1(a)<H_0(a)<0 [/itex].

[itex] H_0(s)<0, H_1(s)<0 [/itex] for all [itex]s\in(a,b] [/itex].

[itex] y_1(s)>y_0(s) [/itex] for all [itex]s \in (a,b] [/itex].

Show:

[itex] H_1(r)<H_0(r) [/itex] for all [itex] r \in (a,b] [/itex].