# Homogeneous or non - homogeneous $?$

The second order differential equation is given by -

$$\frac{d^{2}y}{dx^{2}} + \sin (x+y) = \sin x$$

Is this a homogeneous differential equation $$?$$

Well, I guess this is not a homogeneous differential equation since the form of this equation is not $$a(x)y'' + b(x)y' +c(x)y = 0$$. But the answer is given that it's homogeneous. How can this equation be homogeneous?

Let $$u=x+y$$ ,

Then $$\dfrac{du}{dx}=1+\dfrac{dy}{dx}$$

$$\dfrac{d^2u}{dx^2}=\dfrac{d^2y}{dx^2}$$

$$\therefore\dfrac{d^2u}{dx^2}+\sin u=\sin x$$

This is a inhomogeneous differential equation.

You are correct, as it is not a linear ODE, it is neither homogeneous nor inhomogeneous.

The cited characterization is most likely based on the fact that $$y=0$$ is a solution, but that is only a necessary condition for linearity, not a sufficient one.