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Solution of a differential equation can be written as a sum of its homogenous and particular part: $$y = y_h + y_p,$$ or as a sum of zero-state and zero-input solutions: $$y = y_{ZI} + y_{ZS},$$ where zero-input solution $y_{ZI}$ is the homogenous solution $y_h$ with initial conditions applied, and zero-state solution $y_{ZS}$ is the "full solution" $y_h + y_p$, but with the initial conditions set to zero.

So, $$y = y_h + y_p = y_{ZI} + y_{ZS}.$$

Although it is intuitive, and for $t=0$ we can see the equivalence, I would like to see some real proof that this zero-state/zero-input combination truly holds for every $t$, e.g. is always equal to the "default" solution $y_h + y_p$.

Can someone help me with this?

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This idea is really Duhamels principle, let us apply it to the general first order ODE:

$y'(t)+ay(t)=f(t)$, $y(0)=g$

Multiply by the integrating factor $e^{at}$ we get:

$\frac{d}{dt}(y(t)e^{at})=f(t)e^{at}$, integrating with respect to $t$ we get:

$y(t)e^{at}-y(0)=\int_0^tf(\tau)e^{a\tau}d\tau$

$\Rightarrow y(t)=y(0)e^{-at}+e^{-at}\int_0^tf(\tau)e^{a\tau}d\tau$

$=y(0)e^{-at}+\int_0^tf(\tau)e^{-a(t-\tau)}d\tau=y_c+y_p$,

it is now very clear that $y_c(0)=y(0)$, and $y_p(0)=0$

Now let $S(t)=e^{-at}$ and we get:

$y(t)=y(0)S(t)+\int_0^tf(\tau)S(t-\tau)d\tau=y(0)S(t)+f*S$. (Where $*$ denotes the convolution).

Also this still holds for non constant coefficients, I.e. consider:

$y'(t)+a(t)y(t)=f(t)$, $y(0)=g$ here we take $S(t)=e^{-\int_0^ta(\tau)d\tau}$, and we obtain:

$y(t)=y(0)S(t)+f*S$

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ok.... the theory goes like this..... Classical solution to the constant coefficient Differential equations D(y) + y = X is the sum of both homogenous (complementary) solution and Particular solution. ie y(t) = y(h) + Y(p)

whereas, when we are considering LTI systems, the solution to the Linear system is defined by the sum of its total responses ie the zero input response and zero-state response.... y(t) = y(ZI) + y(ZR)

Also, now remember that in case of linear systems, there are natural and forced responses that are often mistaken to be zero input or zero state response wherein the natural response is the solution of the linear systems that are connected to the characteristic modes of the system.

generally when we are considering to understand the behaviour of an LTI systems the zero input and zero state response method is best because it gives us distinct information on the internal conditions and external inputs whereas, in case of the classical solution it does not give us the advantage of defining the system response to an input x(t) as an explicit function of x(t).

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