[Math] Why do some linear differential equation have no particular solution

Question

For some linear equations, for example:

$$\frac{d^2y}{dx^2} – 3\frac{dy}{dx} – 4y = 0$$

Why is the particular solution 0?

Does this have to do with equations being homogeneous or inhomogeneous?

Answer 1

Yes, it does have to do with the differential equation being homogeneous.

The left side of the differential equation is linear, which has multiple consequences, one of which is that substituting the constant zero function for $y$ gives the result zero.

A homogeneous ODE has zero on the right-hand side of the equation. So $y=0$ is a solution to any homogeneous linear ODE.

So do not say that there is "no particular solution," rather say "the constant zero function is a particular solution", or more briefly, "zero is a particular solution."

This is why homogeneous ODE's are usually easier than non-homogeneous ones.