# Tag Info

## Hot answers tagged differential-equation

Accepted

### Why is particular solution zero for an impulse excitation signal?

The correct form of the statement should be the particular solution is zero for $t>0$ This is simply the case because the input $\delta(t)$ is zero for $t>0$. So for $t>0$, the impulse ...
• 90.5k
Accepted

### When $x(t)$ and the output $y(t)$ are related by linear differential equations, why is the system unstable when $M>N$?

Let's try the simple $N=0$ and $M=1$ case: $$a_0 y(t) = b_0 \frac{d x(t)}{ dt} + b_1 x(t) \tag{1}$$ Now, let's think about stability. A standard approach to stability is that for any bounded (finite-...
• 25.9k
Accepted

• 90.5k
Accepted

• 425
Accepted

### Confusion in initial condition of differential equation using Laplace transform transform

Initial conditions are always given at $t=0^-$, because they define the state of the system before any input is applied, and - by definition - the input is applied at $t=0$. The state at $t=0^+$ is ...
• 90.5k
Accepted

### Stability of a system in time-domain

Solving the characteristic equation $$s^2+\frac32 s-1=0\tag{1}$$ gives the following homogeneous solution: $$y(t)=c_1e^{-2t}+c_2e^{t/2}\tag{2}$$ The requirements of causality and stability are taken ...
• 90.5k
Accepted

### Show that decomposition does not hold for non-linear system

Assume that $y_0(t)$ is the zero-input response. Then $y_0(t)$ must satisfy $$\dot{y}_0(t)+y_0(t)=1\tag{1}$$ because $x(t)=0$. Now let $y_1(t)$ be the zero-state response to an input $x(t)$, ...
• 90.5k
Accepted

• 90.5k
1 vote

### How to solve this degree $2N$ polynomial equation of filter's cut-off frequency?

Since only $\omega$ is unknown, I can see that the equation has the form $x^{2N}a + x^{N}b + xc +d$. Exactly, you're looking for the roots (i.e., points where the value is 0) of a polynomial in $t$. ...
• 31.3k
1 vote
Accepted

### Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?

I have found the following paper named "Finite time differential equations" by V. T. Haimo (1985), where continuous time differential equations with finite-duration solutions are studied, an ...
• 183
1 vote

### Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?

You may be overthinking this. If the input has length $T_x$ and the impulse response of the LTI system has length $T_h$ than the output will have have length $T_x + T_h$. If either input or impulse ...
• 45.6k
1 vote
Accepted

• 28.3k
1 vote

• 90.5k
1 vote

### Difference equation with variable coefficients in MATLAB

When $n=0$ you may need to know $y[-1]$. The current output $y(n)$ depends on the current input $x(n)$ and previous output $y(n-1)$ scaled.
• 669

Only top scored, non community-wiki answers of a minimum length are eligible