Dirac delta in ODE (inhomogenous), where is the discontinuity?Trying to understand where the discontinuity comes from in this equation coming from an ODE course (link at the bottom):x¨+3x˙+2x=δ(t),Initial conditions: x˙(0)=x(0)=0. Assume impulse occurs at t=0+The solution is: x(t)=e−t−e−2t=e−t(1−e−t)And the derivative of the solution evaluated at IC:x˙(t=0)=−e−t+2e−2t=−1+2=1The course mentions that the x(t) is continuous at t=0, but the derivative x˙(t) is not because x˙(0−)=0 whilst x˙(0+)=1. How do I know that x˙(0−)=0? Does one need to read into when impulse occurs to arrive at the conclusion or can it be shown mathematically using the Dirac's delta definition?

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Dirac delta in ODE (inhomogenous), where is the discontinuity?Trying to understand where the discontinuity comes from in this equation coming from an ODE course (link at the bottom):x¨+3x˙+2x=δ(t),Initial conditions: x˙(0)=x(0)=0. Assume impulse occurs at t=0+The solution is: x(t)=e−t−e−2t=e−t(1−e−t)And the derivative of the solution evaluated at IC:x˙(t=0)=−e−t+2e−2t=−1+2=1The course mentions that the x(t) is continuous at t=0, but the derivative x˙(t) is not because x˙(0−)=0 whilst x˙(0+)=1. How do I know that x˙(0−)=0? Does one need to read into when impulse occurs to arrive at the conclusion or can it be shown mathematically using the Dirac&#039;s delta definition?

uvredio0of6 · Accepted Answer

Step 1The theory of distributions is needed to formalise the above, but this is overkill in this context. One can find some intuition by &#039;working under the integral sign&#039;.The net effect of the δ is essentially to introduce a step change in initial conditions.That is, suppose the response of the above system given initial conditions x(0), x(0),x˙(0) is t↦s(x(0),x˙(0);t). Then the response of the system above will be s(0, 0; 0) initially and then s(0, 1; t) for t&amp;gt;0 (note the x˙ term changed to 1).Step 2To see informally how this comes about, integrate the system over a small interval to get∫0ϵx¨(t)dt=x˙(ϵ)−x˙(0)=∫0ϵ−(3x˙(t)+2x(t))dt+∫0ϵδ(t)dtTaking limits we get limϵ↓0(x˙(ϵ)−x˙(0))=1.In particular, the response to the above system, for t&amp;gt;0 is the same as the response of the unforced system to the initial conditions x(0)=0,x˙(0)=1.

Dirac delta in ODE (inhomogenous), where is the

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