Hello. It is a second order linear ordinary differential equation with constant coefficients and initial values.
y(t) = y_{h}(t) + y_{p}(t)
In general one may ask themselves what happened to the transient (general, homogenous) and steady state response (particular, inhomogenous)?
\textbf{Transient Response: Solution Families/Categories Of Finite, Known Cases Of Solutions}
The transient response is the homogenous solution. The homogenous solution occurs for:
- Distinct real roots,
- Repeated real roots,
- Complex roots
\textbf{Steady State response: inhomogenous solution/forcing function }
The transient response is the homogenous solution. The homogenous solution occurs for:
The steady state response is the inhomogeneous solution.
The inhomogeneous solution requires a forcing function. Here, the forcing function is a unit step/Heaviside delayed by pi units.
h(t - \pi)
In general the unit step function scaled and shifted is defined as:
<br />
K h(t - T_{0}) =<br />
\begin{cases}<br />
0 & \text{for } t < T_{0} \\<br />
K & \text{for } t \geq T_{0}<br />
\end{cases}<br />
<br />
Whose derivative is an equally scaled and shifted delta pulse:
<br />
\frac{d}{dt} (K h(t - T_{0})) = K \delta(t - T_{0})<br />
<br />
The derivative of the dirac delta (an term included within and of the second derivative of y(t)) is a distribution defined over a test function phi:
<br />
\delta'(t) = \int_{-\infty}^{\infty} \delta'(t) \phi(t) \, dt = -\phi'(0)<br />
<br />
For O.D.E, the transient response is the homogenous solution to the equation.
The steady state response is directly related to the forcing function.
it is fair to say that the transient response of a second-order linear differential equation is both differentiable everywhere, continuous in all cases of roots:
- real and distinct,
-real and repeated,
- and complex
Hence y(t) would be discontinuous (jump discontinuity)...and the nature of the y'(t) and y"(t) follows from the jump discontinuity of the step function, a term in y(t)...
....
Discontinuities in y(t) occur at the point where the unit step/heaviside function is applied
So for the solution for this family/category of these equations as the behavior of the homogenous parts are well defined across all finite and known cases, the discontinuity behavior arises from the forcing function, inhomogeneous solution.