# How do you determine the properties of a differential equation?

If I was given a differential equation of the form $$y'(t) + a(t)y(t) = x(t)$$ how would I be able to decide its linearity, time-invariance, and causality?

• well, it ain't time-invariant if $a(t)$ changes. if $a(t)=a$ is constant, it is time-invariant. – robert bristow-johnson Nov 14 '18 at 3:19
• Thank you! Appreciate it. Apologies for the formatting in my question. – Benny Nov 14 '18 at 3:21
• i haven't been picking on you about formatting. sometimes it's easier just to do it. – robert bristow-johnson Nov 14 '18 at 3:23
• usually we use the terms "linearity", "time-invariancy", and "causality" to describe properties of a system with an input or inputs and an output or outputs. that system may have an input/output relationship defined by a differential equations such as you have shown. you might want to be explicit about what signal is the input and what is the output. – robert bristow-johnson Nov 14 '18 at 3:27
• without stating the initial conditions as initial rest you cannot determine whether the equation signifies an LTI systen or not. – Fat32 Nov 19 '18 at 11:05

If a system is represented by a differential equation then it must be LINEAR. If the coefficients of differential equation are function of time then it is time variant otherwise time invariance. For causality and stability we will need its transfer function for any comment.

If the input $$x(t)$$ and the output $$y(t)$$ and their derivatives occur linearly in the equation, i.e., not in the form $$f\left(x^{(n)}(t)\right)$$ or $$f\left(y^{(n)}(t)\right)$$, where $$f(\cdot)$$ is a non-linear function, then the differential equation can describe a linear system.

Time-invariance is determined by the factors with which $$x(t)$$ and $$y(t)$$ and their derivatives are multiplied. If they are constant, and, consequently, not functions of $$t$$, then the equation can describe a time-invariant system. If that is the case, and if the equation is also linear, then it is called Linear Constant-Coefficient Differential Equation. The general form of such an equation is

$$\sum_{n=0}^Na_n\frac{d^ny(t)}{dt^n}=\sum_{n=0}^Mb_n\frac{d^nx(t)}{dt^n}\tag{1}$$

Given only the differential equation, you cannot draw any conclusions concerning causality. Taking your example with $$a(t)=a$$ being constant (for time-invariance), you can verify that the given differential equation describes two different systems. One is a causal system with impulse response $$h(t)=e^{-at}u(t)$$, and the other one is anti-causal with impulse response $$h(t)=-e^{-at}u(-t)$$.

• for your first paragraph; that describe a linear differential equation but necessarily a linear system, unless initial conditions are all zero (initial rest). Specifically if $y(0)=1$ for then the output will not be linear in $x$. That's also true for time-invariance; initial conditions should be consistent. Assuming initial rest is an easy cover up... – Fat32 Nov 19 '18 at 11:12
• @Fat32: yes, I agree. I've had many (fruitless) discussions about this, and I wanted to avoid it this time. This is why I said "can describe a linear system". – Matt L. Nov 19 '18 at 11:16
• And I've added a second "can" in the other paragraph on time-invariance. – Matt L. Nov 19 '18 at 11:18
• Anyway, Benny doesn't seem to care anymore ... :) – Matt L. Nov 19 '18 at 11:20