Where Does the δ(t) Term Come From in First Order Unit Step Response?

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SUMMARY

The δ(t) term in the first order unit step response equation x' + kx = (r/k + c2)δ(t) represents an impulse function that is crucial for understanding system behavior at t=0. This term evaluates to zero for t≠0, indicating that it only influences the system at the exact moment of the impulse. The confusion arises from the interpretation of δ(t) as it does not evaluate to (r/k + c2) outside of t=0, but rather acts as a mathematical tool to model instantaneous changes in the system.

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  • Understanding of differential equations, specifically first-order linear equations.
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  • Study the properties of the Dirac delta function in detail.
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charlie.elvers
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I think I mostly understand how this works, and it makes intuitive sense. However, I'm a little bit confused by one step in the proving of this.

On the second page of the attached PDF, there is this statement:

x' + kx = (r/k + c2)δ(t) ... (followed by the cases of t<0 and t>0

What comes after δ(t) is easy for me to follow, but I don't know where this δ(t) term comes from. Does this term evaluate to 0? If so, how? I would think it'd evaluate to (r/k + c2) because the (r/k + c2) is independent of t

Thanks.
 

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Sorry folks. I just read the forum rules, and I think this is probably the wrong forum for such a thread.
 

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