# 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 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