Definition of Homogenous function:
A function $f(x)$ is said to be homogeneous of degree $n$ if, by introducing a constant parameter $\lambda$, replacing the variable $ x $ with $\lambda x$ we find: $f(\lambda x)$ = $\lambda^n f(x)\,.$ This definition can be generalized to functions of more-than-one variables; for example, a function of two variables $f(x,y)$ is said to be homogeneous of degree $ n $ if we replace both variables $x$ and $y$ by $ \lambda x $ and $ \lambda y$, we find: $f(\lambda x, \lambda y) = \lambda^n f(x,y)\,.$
My book says that $dy/dt=sin(2t)y$ is homogenous. But isn't
$Q(t,y)dy-P(t,y)dt=0$ where $Q(t,y)=1$ and $P(t,y)=sin(2t)y$
And functions $Q$ and $P$ are not of the same order so the $dt$ and $dy$ coefficients are not of same order thus the Diff eqn is not homogenous ?
Does the same logic apply to
$AB(dy(x)/dx) + y(x)=0$ where A and B are constants ? Also, Both eqns are linear right ?
Sometimes the term homogeneous is used to mean that if $y$ is a solution of the differential equation, so is $\lambda\cdot y$ for any constant $\lambda$.
Unfortunately, homogeneous has several different meanings and you will have to try to figure out which one is meant in this context...