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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?

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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.

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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.

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